Published in Teor. Mat. Fiz. 89 (1991) 56–72 [Theor. Math. Phys. 89 (1991) 1052–1064] A method of evaluating massive Feynman integrals Institute for Nuclear Physics, Moscow State University, A general method is proposed for calculating massive Feynman integrals on the basis of a representation of the massive denominators in the form of Mellin–Barnesintegrals. This method is used to obtain expressions for some classes of single-loopmassive Feynman integrals of propagator and vertex type (for arbitrary values ofthe powers of the denominators and dimension of space). The results are presentedin the form of hypergeometric functions, making it possible to investigate differentranges of variation of the momenta.
Because of the need to make various calculations in gauge theories (QCD, electroweakmodel, etc.), it is very important to develop methods that permit exact calculation ofvarious types of Feynman diagrams containing both massless and massive particles. Thisis true, in particular, for calculation of interaction cross sections and decay widths invarious orders of perturbation theory, for the investigation of the coefficient functions inoperator expansions, for the renormalization-group analysis of β functions, anomalousdimensions and invariant charges, for study of the behaviour of Green’s functions, theproblem of anomalies, etc. In many cases it is most convenient for the calculation ofthe corresponding Feynman integrals to use dimensional regularization [1, 2] (see also thereview [3]), the use of which makes it possible, in particular, to preserve gauge invarianceat all stages.
At the present time, the greatest successes have been achieved in the development of methods of calculation of massless Feynman integrals of propagator type (i.e., dependenton a single external momentum): the method of Gegenbauer polynomials [4, 5], inte-gration by parts [6], the uniqueness method (see, for example, [7]), and also some othermethods [8, 9, 10] (see also the review [11]). Massless integrals of vertex type (with twoindependent external momenta) have a much more complicated structure, and some caseswere investigated in [12, 13, 14].
At the same time, for the calculation of many processes, especially those including heavy particles, one cannot avoid the use of Feynman integrals with massive denomina-tors. However, at the present time not many exact expressions are known for differentdimensionally regulated massive Feynman integrals (see, for example, [1, 15, 16, 17] andthe references therein).
In the present paper, we propose a general method for obtaining exact solutions for Feynman integrals containing massive denominators. The idea of the method is to rep-resent the massive denominators in the form of Mellin–Barnes integrals with subsequentcalculation of the corresponding massless integrals. This method makes it possible toobtain results for arbitrary values of the space-time dimension n and indices of the lines(degrees of the corresponding denominators), making it possible to use the obtained ex-pressions in both dimensional and analytic regularization. These results can be expressedin terms of functions of hypergeometric type, so that different ranges of variation of themomenta can be investigated (see below). We note that we can, without loss of general-ity, consider only scalar integrals, since all integrals with Lorentz tensor structure in thenumerator can be reduced to scalar integrals, by using formulae of the type given, forexample, in [18, 19].
The paper is organized as follows. Section 1 is devoted to discussion of the general idea of the proposed method, which can be used for arbitrary (including multi-loop)massive integrals. In Section 2, the general technique is illustrated by the example of thecalculation of massive Feynman integrals of propagator type. In Section 3, we calculatesome classes of massive integrals of vertex type. In the Conclusions, we formulate anddiscuss the main results of the paper.
Suppose we have a Feynman integral that contains one or several massive denominatorsof the form where k is the momentum of the corresponding line, m is the mass (for different lines, themasses may be different), and β is the index of the line (the degree of the denominator).
In addition, the integral can also have massless denominators. The infinitesimally smalladdition (+i0) determines the usual “causal” way of dealing with the singularities in thepseudo-Euclidean space. In what follows, we shall assume that all squares of the momentain the denominators have such additions (including those in the employed expansions andintegral representations), and we shall not write them out explicitly.
As is well known, the direct calculation of such massive integrals using standard meth- ods (α-representation or Feynman parameters) involves great difficulties in the calculationof the parametric integrals, and general expressions have been obtained only for the sim-plest cases. There is another procedure for obtaining asymptotic expansions of suchintegrals with respect to parameters of the type m2/p2 (p is an external momentum)associated with the use of the R∗ operation (see [20, 21, 22]).
We expand the considered denominator (1) in a series in m2/k2: If we now formally insert this expansion in the integrand, we obtain an infinite sum ofintegrals in which the massive denominator (1) is replaced by massless denominators.
It is obvious that if all the massive denominators in the considered Feynman integralare expanded in this manner, then it will be reduced to the sum of the correspondingmassless integrals. In particular, if there is just one external momentum p, then weobtain in this way an expansion in m2/p2. However, it is easy to see in simple examplesthat the obtained result will be incorrect. This is because the expansion (2) is valid onlyin the region |k2| > m2, while the integration is over all k, including the region in which|k2| < m2. In this region, the expansion (2) must, in general, be replaced by a differentone: Thus, the correct expansion of the denominator (1) is a combination of (2) and (3) withcorresponding θ functions (for further work with such expressions, it is better to useEuclidean variables). However, the presence of the θ functions greatly complicates the corresponding massless integrals, and the entire gain from the reduction of the massiveintegrals to the massless ones is almost lost. One of the ways out of the resulting situationis to use the procedure proposed in [20, 21, 22]. It is as follows. Suppose we use fordenominators of the form (1) the expansion (2) and compensate the “incorrectness” ofthis expansion in the region |k2| < m2 by appropriately chosen counterterms (a generalprescription for the construction of such counterterms is given). This makes it possible toobtain correct asymptotic expansions of the corresponding integrals in powers of m2/p2.
We propose a different method for calculating massive Feynman integrals. The idea of the method is to use the Mellin–Barnes representation for the function 1F0, where the contour in the complex plane of s separates the “left” series of poles of Γfunctions in the integrand from the “right” poles (in what follows, all such integralswill be understood in this sense). To calculate the integral (4), we can use the residuetheorem, closing the contour at infinity in the right or left half-plane in order to make theintegrand decrease (depending on the value of |z|). For example, for |z| < 1 we must closethe contour in (4) on the right, and for |z| > 1 on the left (and the obtained expressionis equal to the sum over the residues of Γ(−s) or Γ(β + s), respectively). In this manner,we obtain the well-known expressions for the analytic continuation of the hypergeometricfunctions (see, for example, [23]). Thus, the main formula of the method is (we repeat that all squares of momenta contain infinitesimally small imaginary additions:k2 ↔ k2 +i0). The advantage of this method is that formula (5) contains both (2) and (3):for |k2| > m2, we obtain (summing over the residues of Γ(−s)) the expansion (2), and for|k2| < m2 (summing over the residues of Γ(β + s)) the expansion (3). At the same time,we can use the ordinary expressions for the massless integrals, replacing the correspondingindex β by (β + s). The use of this method also has a number of other helpful properties,which will be noted below in the calculation of definite classes of integrals.
It should be noted that the appropriateness of using the Mellin–Barnes representation for the hypergeometric functions (and also the Mellin transform) in the calculation of one-dimensional integrals has already been noted (see, for example, [24, 25]). In particular, itwas used in [26, 19, 13] to study parametric integrals that arise when the α-representationis used to calculate certain Feynman integrals. In particular, the Mellin transform wasused in [27] to analyze α-parametrized integrals in the investigation of singularities andthe asymptotic behaviour of massive Feynman amplitudes. Note also that in [28] a studywas made of some aspects of the calculation of massive integrals of propagator type byusing a single Mellin transformation with respect to the square of the external momentumand considering the Mellin transforms of such integrals. Our proposed technique of theMellin–Barnes representation directly for the massive denominators differs from theseapproaches, and from our point of view is more convenient for calculating definite classesof massive Feynman integrals.
We now consider examples of the use of the basic formula of the method (5) for integralsof propagator type (containing one external momentum). In this section, we shall operatewith single-loop integrals of the form where n = 4 − 2ε is the spacetime dimension (in the framework of dimensional regulariza-tion [1, 2]). A special case of the integral (6) with α = β = 1 was considered in particularin [15].
We give first a detailed treatment of a well-known simple example: m1 = 0, m2 ≡ m.
ds (−m2)s Γ(−s) Γ(β + s) J(0)(α, β + s) , where the symbol J(0) denotes the corresponding massless integral, the result for whichis well known: Γ(n/2 − α) Γ(n/2 − β) Γ(α + β − n/2) Substituting (8) in (7) and replacing the variable of integration s by (n/2 − α − β − s)(replacements of such type do not violate the condition for separating by a contour theright and left series of poles — all that happens is that “right” and “left” change places),we have J(α, β; 0, m) = πn/2 i1−n (−m2)n/2−α−β Γ(α) Γ(β) s Γ(−s) Γ(α + s) Γ(α + β − n/2 + s) Here and in what follows, the phase is defined in such a way that i1−n(−m2)n/2 = i(m2)n/2.
Closing the contour of integration on the right, we obtain J(α, β; 0, m) = πn/2 i1−n (−m2)n/2−α−β Γ(α) Γ(β) where 2F1 is Gauss’ hypergeometric function. If we close the contour of integration in (9)on the left, we obtain a result in the form of functions of m2/p2: J(α, β; 0, m) = πn/2 i1−n (p2)n/2−α−β Γ(n/2 − α) Γ(n/2 − β) Γ(α + β − n/2) Of course, the same result can be obtained from (10) by applying the well-known formulaof analytic continuation of the function 2F1 (see, for example, [23]). Note also that otherformulae of analytic continuation of 2F1 [23] make it possible to express the obtained resultin the variables (m2 − p2)/m2 and (p2 − m2)/p2 (and their inverses). Such expansions areof interest for investigation of the behaviour of the Green’s functions near the mass shell.
In particular, going over to the variable (p2 − m2)/p2, we reproduce the result obtainedin [29] by means of the α-representation.
Returning to the expression (11), we note that on formal substitution of the expansion (2) we would obtain only the first function 2F1 in the braces, and the result would beincorrect. Application of the procedure [20, 21, 22] here reduces to term-by-term recoveryof the expansion coefficients of the second function 2F1, which in our approach is obtainedautomatically.
Note also that taking the limit α → 0 in (10) gives J(0, β; 0, m) = πn/2 i1−n (−m2)n/2−β This result agrees with the well-known result of [1]. By means of the proposed method,the expression (12) can also be obtained directly by using the property [9] We now consider another interesting special case of the integral (6), when m1 = m2 ≡ Applying the general formula (5) twice to the denominator of (13), and using (8), weobtain J(α, β; m, m) = πn/2 i1−n (p2)n/2−α−β [Γ(α) Γ(β)]−1 Γ(n/2 − α − s) Γ(n/2 − β − t) Γ(α + β − n/2 + s + t) Making the change of variable t = n/2 − α − β − s − u, and also using Barnes lemma tocalculate the integral over s (see, for example, [24, 23]), J(α, β; m, m) = πn/2 i1−n (−m2)n/2−α−β [Γ(α) Γ(β)]−1 u Γ(−u)Γ(α+u)Γ(β+u)Γ(α+β−n/2+u) Hence, closing the contour of integration to the right or to the left and using the well-known duplication formula for the Γ function, Γ(2z) = 22z−1π−1/2Γ(z)Γ(z + 1/2), weobtain J(α, β; m, m) = πn/2 i1−n (−m2)n/2−α−β Γ(n/2 − α) Γ(n/2 − β) Γ(α + β − n/2) J(α, β; m, m) = πn/2 i1−n (p2)n/2−α−β α + β − n/2, (α + β − n + 1)/2, (α + β − n + 2)/2 4m2 β, (β − α + 1)/2, (β − α + 2)/2 4m2 α, (α − β + 1)/2, (α − β + 2)/2 4m2 (with regard to the hypergeometric functions encountered in the paper, including 3F2,see the Appendix). Note that, expanding (17) with respect to ε = (4 − n)/2 in the caseα = β = 1, we obtain the well-known result (see, for example, [30]) expressed in terms ofelementary functions.
Finally, we consider the general case of the integral (6). Double application of formula (5) (with allowance for the expression (8)) gives J(α, β; m1, m2) = πn/2 i1−n (p2)n/2−α−β [Γ(α) Γ(β)]−1 Γ(n/2 − α − s) Γ(n/2 − β − t) Γ(α + β − n/2 + s + t) Calculating the contour integrals, we obtain J(α, β; m1, m2) = πn/2 i1−n (−m22)n/2−α−β ×F4(α, α + β − n/2; n/2, α − n/2 + 1| p2/m2, m2/m2 where F4 is Appell’s hypergeometric function of two variables [31, 23] (see also the Ap-pendix). If we consider the case m2 < m1, then in the expression (20) we should in-terchange (m1, α) ↔ (m2, β). From (19), we can also obtain the result in terms of thevariables m2/p2 Γ(n/2 − α) Γ(n/2 − β) Γ(α + β − n/2) J(α, β; m1, m2) = πn/2 i1−n (p2)n/2−α−β × F4(α + β − n/2, α + β − n + 1; α − n/2 + 1, β − n/2 + 1| m2/p2, m2/p2 4(β, β −n/2+1; n/2−α+1, β −n/2+1| m21 4(α, α−n/2 + 1; α−n/2+1, n/2−β +1| m21 Thus, for the general integral (6) we have constructed the representation (19), from whichwe can obtain the result in the form of hypergeometric functions for different relationsbetween the masses m1 and m2 and the momentum p (for example (20) and (21)).
Note that from the expression (20) for α = β = 1 and p2 = 0 we obtain the well-known In this section, we consider examples of the application of the proposed technique tosingle-loop “triangle” integrals of vertex type (with two independent external momenta)containing massive denominators (see Fig. 1).
It is obvious that application of formula (5) to massive denominators requires infor- mation about the corresponding massless integrals: (as before, we understand the causal way of dealing with the singularities in the pseudo-Euclidean space). A general expression for the integrals (22) was obtained in [13] and canbe represented in the form J(0)(µ, ν, ρ) = πn/2 i1−n [Γ(µ) Γ(ν) Γ(ρ) Γ(n − µ − ν − ρ)]−1 × (k2)n/2−µ−ν−ρ Γ(µ) Γ(µ+ν +ρ−n/2) Γ(n/2−µ−ν) Γ(n/2−µ−ρ) ×F4(µ, µ + ν + ρ − n/2; µ + ν − n/2 + 1, µ + ρ − n/2 + 1| p2/k2, q2/k2)+(q2)n/2−µ−ρ(k2)−ν Γ(ν) Γ(n/2 − ρ) Γ(n/2 − µ − ν) Γ(µ + ρ − n/2)×F4(ν, n/2 − ρ; µ + ν − n/2 + 1, n/2 − µ − ρ + 1| p2/k2, q2/k2)+(p2)n/2−µ−ν(k2)−ρ Γ(ρ) Γ(n/2 − ν) Γ(µ + ν − n/2) Γ(n/2 − µ − ρ)×F4(ρ, n/2 − ν; n/2 − µ − ν + 1, µ + ρ − n/2 + 1| p2/k2, q2/k2)+(p2)n/2−µ−ν(q2)n/2−µ−ρ(k2)µ−n/2×Γ(n − µ − ν − ρ) Γ(n/2 − µ) Γ(µ + ν − n/2) Γ(µ + ρ − n/2) ×F4(n−µ−ν −ρ, n/2−µ; n/2−µ−ν +1, n/2−µ−ρ+1| p2/k2, q2/k2) , where k = q − p, and F4 (as in (20) and (21)) is Appell’s hypergeometric function oftwo variables (see (A.2)). In particular, if one of the parameters µ, ν, ρ is zero, then in(23) there remains, respectively, only the first, second, or third term in the braces and weobtain the well-known result (8), whereas for µ+ν +ρ = n only the fourth term “survives”,and we obtain the uniqueness relation (see, for example, [7]). For our purposes, it will beconvenient to use the representation J(0)(µ, ν, ρ) = πn/2 i1−n (k2)n/2−µ−ν−ρ [Γ(µ) Γ(ν) Γ(ρ) Γ(n − µ − ν − ρ)]−1 ×Γ(n/2 − µ − ν − s) Γ(n/2 − µ − ρ − t)×Γ(µ + s + t) Γ(µ + ν + ρ − n/2 + s + t) , from which we can readily obtain both (23) and the corresponding expressions in termsof functions of other dimensionless momentum variables. We note that representations ofsuch type were used in some special cases (for n = 4) in [26].
We now consider vertex integrals with one massive denominator, (r2 − m2)µ [(p − r)2]ν [(q − r)2]ρ Use of the basic formula (5) of the method gives du (−m2)u Γ(−u) Γ(µ + u) J(0)(µ + u, ν, ρ) .
Substituting here the representation (24) and going over from the variable u to(n/2 − µ − ν − ρ − s − t − u), we obtain J1(µ, ν, ρ; m) = πn/2 i1−n (−m2)n/2−µ−ν−ρ [Γ(µ) Γ(ν) Γ(ρ)]−1 ×Γ(−s) Γ(−t) Γ(−u) Γ(n/2 − ν − ρ − u) Γ(µ + ν + ρ − n/2 + s + t + u) Γ(ν + s + u) Γ(ρ + t + u) Note that here we have two series of poles in the right half-plane of the variable u (dueto Γ(−u) and Γ(n/2 − ν − ρ − u)). Calculating the integrals (26), we obtain the result Γ(µ + ν + ρ − n/2) Γ(n/2 − ν − ρ) J1(µ, ν, ρ; m) = πn/2 i1−n (−m2)n/2−µ−ν−ρ k2 n/2−ν−ρ Γ(n/2 − ν) Γ(n/2 − ρ) Γ(ν + ρ − n/2) n − ν − ρ; n/2 − ν − ρ + 1 m2 m2 where Φ1 is a function of hypergeometric type that can be expressed in terms of Lauri-cella’s generalized function of three variables (see (A.4)): 1 : 1, 1, 1), (a2 : 1, 0, 1), (a3 : 0, 1, 1) where (a)j ≡ Γ(a + j)/Γ(a) is the Pochhammer symbol. Note that the general formula(26) makes it possible to go over to other dimensionless variables (for example, m2/p2,m2/q2, etc.). Note also that for µ = 0 formula (27) corresponds to the well-known result(8), as it should.
It is sometimes of interest to consider symmetric deviation from the mass shell with respect to two legs of the corresponding Feynman diagram, q2 = p2 (see, for example,[19, 13]). Then the function Φ1 can be represented as a generalized hypergeometricfunction of two variables: 1 : 1, 1), (a2 +a3 : 1, 2) : (a2 : 1), (a3 : 1) Because the sum over j in (29) represents the function 2F1, we can here (as in the ex-pression (10)) readily obtain, by means of the formulae of analytic continuation [23],expansions with respect to the variables 1/z, (1 − z), (z − 1)/z, etc., which are often usedto investigate asymptotic behaviour in different regions.
We now consider a vertex integral with two massive denominators and the same mass (r2)µ [(p − r)2 − m2]ν [(q − r)2 − m2]ρ .
Applying (5) to the massive denominators yields ×Γ(−v) Γ(−w) Γ(ν + v) Γ(ρ + w) J(0)(µ, ν + v, ρ + w) .
Substituting, further, the expression for J(0) (24), going over to the variable u by meansof the substitution w = n/2 − µ − ν − ρ − s − t − v − u, and calculating the integral overv by means of Barnes lemma (15), we obtain J2(µ, ν, ρ; m) = πn/2 i1−n (−m2)n/2−µ−ν−ρ [Γ(µ) Γ(ν) Γ(ρ)]−1 Γ(µ + ν + ρ − n/2 + s + t + u) Γ(µ + s + t) Γ(ν + s + u) Γ(ρ + t + u) Γ(n/2 − µ + u) Hence, closing the contours with respect to the variables s, t, and u on the right, we canobtain the expression J2(µ, ν, ρ; m) = πn/2 i1−n (−m2)n/2−µ−ν−ρ µ + ν + ρ − n/2, µ, ν, ρ; n/2 − µ where the function Φ2 can also be expressed in terms of Lauricella’s generalized functionof three variables (see (A.4)): 1 : 1, 1, 1), (a2 : 1, 1, 0), (a3 : 1, 0, 1), (a4 : 0, 1, 1) : (b : 1) In the case of symmetric deviation from the mass shell (q2 = p2), and bearing in mindthat in our case c2 = a3 + a4 = ν + ρ, we can represent Φ2 in terms of Kampe de Ferietfunction (see (A.3)) where we have also used the duplication formula for the Γ function. As in the case (29),the sum over j represents 2F1 and can be analytically continued to other variables.
Finally, we consider a vertex integral with three massive denominators when all the (r2 − m2)µ [(p − r)2 − m2]ν [(q − r)2 − m2]ρ .
Using formula (5), we can express it in terms of the already considered integral J2, Substituting here the representation (31) and again using Barnes lemma (15), we obtainthe expression J3(µ, ν, ρ; m) = πn/2 i1−n (−m2)n/2−µ−ν−ρ [Γ(µ) Γ(ν) Γ(ρ)]−1 Γ(µ + ν + ρ − n/2 + s + t + u) Γ(µ + s + t) Γ(ν + s + u) Γ(ρ + t + u) Hence, proceeding as in the previous cases, we obtain the symmetric result J3(µ, ν, ρ; m) = πn/2 i1−n (−m2)n/2−µ−ν−ρ where Φ3 can also be represented in terms of a generalized hypergeometric function ofthree variables (see (A.4)): 1 : 1, 1, 1), (a2 : 1, 1, 0), (a3 : 1, 0, 1), (a4 : 0, 1, 1) 1 : 1, 1, 1), (a2 : 1, 1, 0), (a3 : 1, 0, 1), (a4 : 0, 1, 1) Here we have also used the duplication formula for the Γ function. In particular, whenp2 = q2 = k2 = 0, we obtain the well-known result (12) (with β = µ + ν + ρ).
If we consider the case q2 = p2 (z1 = z2 ≡ z), then we readily obtain the result in the form of a generalized hypergeometric function of two variables (see (A.4)), 1 : 1, 1), (a3 + a4 : 1, 2) : (a2 : 1); (a3 : 1), (a4 : 1) (c/2 : 1, 1), ((c + 1)/2 : 1, 1); (a3 + a4 : 2) Note that reduction formulae of the type (29), (34) and (39) can be readily obtained fromthe corresponding Mellin–Barnes representations (26), (31) and (36) by means of Barneslemma (15).
Consider a simple special example of application of the general formula (37). If p2 = q2 = 0, then from (37) and (38) (or from (39)) we find that Such integrals are needed, in particular, in the calculation of the diagram correspondingto the Higgs boson production via the gluon fusion, through a heavy-quark loop (see, forexample, [32]).
For example, for the specific integral with µ = ν = ρ = 1 we take the limit n → 4, Using the formulae given in [33], we obtain Concluding this section, we note that we have represented the results for the integrals(25), (30) and (35) in the form of hypergeometric functions of the variables p2/m2, q2/m2,and k2/m2, equations (27), (32) and (37), since in these variables the obtained expressionstake their most compact form. Expansions in terms of other variables can be obtainedfrom the general representations (26), (31) and (36).
In this paper, we have considered a general method of calculating Feynman integrals thatcontain massive denominators. The Mellin–Barnes representation (5) enables us to reducethe massive Feynman integrals to massless ones. At the same time, in contrast to theexpansions in the series (2) and (3), the representation (5) is true for all relations betweenthe momentum and the mass. If we know an expression for the massless integral witharbitrary index of the line corresponding to the massive denominator, then the massiveintegral can also be calculated. Note that we have calculated the integrals in pseudo-Euclidean space; however, it is clear that the transition to the Euclidean case does notpresent difficulties.
The method makes it possible to calculate the integrals for arbitrary line indices and dimension of space, and therefore it can be used in both dimensional and analytic regu-larization schemes. In particular, this makes it possible to express by a single formula allresults for the class of Feynman integrals in the most interesting case of integer indices(in practice, such integrals can be calculated recursively). In addition, expressions forintegrals with arbitrary line indices can be used in an investigation of the compatibility ofpower-like solutions with dynamical integral equations for Green’s functions (for example,in investigation of the infrared behaviour of quantum chromodynamics). Note also thatthe method may be helpful in the case when massless singularities are regulated by theintroduction of a small mass.
As a rule, the obtained expressions for the integrals can be represented in the form of hypergeometric functions of dimensionless combinations of squares of the momenta andmasses. This is extremely helpful, since, using the formulae of analytic continuation, itis possible to go over from certain variables to others and investigate different ranges ofvariation of the momenta. In particular, to investigate processes with heavy particlesit is convenient to use functions of arguments of the form p2/m2, and for light particlesfunctions of arguments of the form m2/p2. It is also possible to investigate regions near themass shells of the particles, and also the behaviour near threshold values of the momenta.
In the present paper, we have illustrated the application of the proposed method for the example of classes of single-loop massive Feynman integrals of propagator and vertextype. As far as we know, some of the results have been obtained for the first time. Fordefinite (integer) values of the powers of the denominators, and also after expansion withrespect to ε = (4 − n)/2, the general formulae simplify considerably; at the same time, it is convenient to use the formulae given in the handbook [33]. Great simplifications arealso achieved by various extra conditions (for example, vanishing of some line index orsquare of a momentum, treatment of certain momenta on the mass shell, etc.). For theknown limiting cases of such kind, the obtained formulae give the correct results.
It is clear that the considered examples by no means exhaust the results that can be obtained by the proposed method. In particular, it can be used to calculate many-loop integrals with massive denominators, vertex integrals with larger number of externallines, integrals in the axial gauge, etc. We hope that continuation of investigations in thisdirection will make it possible to increase the number of exactly calculable diagrams inquantum field theory.
We thank B. A. Arbuzov and V. I. Savrin for interest in the work and support, and also K. G. Chetyrkin, V. A. Ilyin, A. L. Kataev, S. A. Larin, A. A. Pivovarov, V. A. Smirnov,F. V. Tkachov and N. I. Ussyukina for helpful discussions and critical comments.
In this Appendix, we give definitions of the hypergeometric functions occurring in thepresent work (more detailed information about these functions can be found, for example,in [23, 24, 31, 34, 35]). Note that expansions of these functions in other regions of variationof the variables can be obtained by means of analytic continuation (for this, it is convenientto represent the corresponding functions in the form of Mellin–Barnes integrals).
The generalized hypergeometric function of one variable is defined by where (a)j ≡ Γ(a + j)/Γ(a) is the Pochhammer symbol.
Appell’s hypergeometric function of two variables F4 has the form A more general hypergeometric function of two variables (Kampe de Feriet function) 1, . . . , cC : d1, . . . , dD; d , . . . , d For example, it is readily seen that F4 = F 2:0;0 Finally, the generalized Lauricella function of N variables [35] has the form [a : α(1), . . . , α(N)] : [b(1) : β(1)]; . . . ; [b(N) : β(N)] [c : γ(1), . . . , γ(N)] : [d(1) : δ(1)]; . . . ; [d(N) : δ(N)] [a : α(1), . . . , α(N)] ≡ (a1 : α(1) [c : γ(1), . . . , γ(N)] ≡ (c1 : γ(1) In (A.4), it is understood that all α, β, γ, δ are non-negative integers, although this for-mula can be generalized to all non-negative values of these parameters, if all the Pochham-mer symbols are represented in terms of the corresponding Γ functions (see, for example,[34, 35]). Note that for N = 2 the function (A.4) is sometimes called the generalizedKampe de Feriet function.
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