Coordinate space calculation of two- and three-loop sunrise-type diagrams, elliptic functions and truncated Bessel integral identities

We integrate three-loop sunrise-type vacuum diagrams in $D_0=4$ dimensions with four different masses using configuration space techniques. The finite parts of our results are in numerical agreement with corresponding three-loop calculations in momentum space. Using some of the closed form results of the momentum space calculation we arrive at new integral identities involving truncated integrals of products of Bessel functions. For the non-degenerate finite two-loop sunrise-type vacuum diagram in $D_0=2$ dimensions we make use of the known closed form $p$-space result to express the moment of a product of three Bessel functions in terms of a sum of Claussen polylogarithms. Using results for the nondegenerate two-loop sunrise diagram from the literature in $D_0=2$ dimensions we obtain a Bessel function integral identity in terms of elliptic functions.


Introduction
The computation of higher order vacuum diagrams in quantum field theory has been extended to ever higher loop levels involving different degrees of mass degeneracy and/or zero mass values for their mass configurations. The multi-loop calculations have traditionally been carried out in momentum space (p-space). For the sunrise-type subclass of these higher order vacuum diagrams it is much simpler to integrate the diagrams using configuration-space (x-space) techniques. The comparison of the results of the two calculations will provide a welcome nontrivial cross check on the correctness of the respective p-and x-space calculations in as much as a subclass of the vacuum diagrams are of sunrise-type. In addition, if one can avail of closed form p-space results, one arrives at new nontrivial integral identities involving moments of Bessel functions. In the case of the two-loop nondegenerate sunrise diagram this leads to an identity of moments of Bessel functions in terms of elliptic integrals. We mention that Bessel function integral identities involving elliptic functions have been studied before in Refs. [1,2]. This paper was triggered by the recent appearance of two p-space calculations of the nondegenerate three-loop vacuum diagrams (all four masses different). We numerically confirm the results of the vacuum sunrise-type diagrams in Refs. [3,4] using x-space techniques. In Sec. 2 we briefly review some general features of the x-space approach and describe the splitting technique used to separate singular (analytical) and finite (numerical) parts in D = D 0 − 2ε spacetime dimensions where D 0 = 4 in the present application.
The particular form of the splitting technique preserves the inherent symmetry of the nondegenerate multiloop vacuum and sunrise integrals w.r.t. the exchange of different rungs or masses in the diagrams. This leads to the notion of truncated Bessel integrals. In Sec. 3 we use closed form results for the finite parts of the three-loop vacuum diagrams from the literature to obtain new integral identities for a set of truncated Bessel integrals.
In Sec. 4 we compare results for the nondegenerate two-loop sunrise diagram with results from Refs. [5,6] for D 0 = 2 to obtain again an integral identity. Our conclusions are found in Sec. 5.

Splitting technique for Bessel integrals
The configuration space calculation of n-loop sunrise-type diagrams with N = n + 1 different masses in arbitrary spacetime dimensions has been studied by us in a series of papers [7,8,9,10,11,12,13,14]. The starting point is the central identity for the p-space correlator functionΠ(p) given bỹ where p = √ p 2 and x = √ x 2 are the absolute values of the four-vectors of four-momentum and spacetime, respectively. The relation of the spacetime dimension to the parameter λ is given by D = 2λ + 2. n-loop sunrise-type integrals are UV divergent for nD ≥ 2(n + 1) (or nλ > 1). In order to parametize the singularities, we use dimensional regularization by is the Bessel function of the first kind, and is the free propagator in (Euclidean) spacetime for a particle line with mass m. The function K λ (mx) denotes the Bessel function of the second kind, also known as the McDonald function. In the case that the particle mass vanishes (m = 0), the propagator simplifies to since the corresponding Bessel function of the second kind is replaced by a simple power dependence. A corresponding simplification occurs in the case of a vacuum diagram ( Two additional features can easily be implemented in the configuration space calculus.
Higher powers of propagators can be inserted by calculating the derivative with respect to the corresponding squared mass, i.e.
which leads to A more complicated task is to implement the extension to tensor integrals. One then has to expand the exponential factor of the Fourier transform in terms of Gegenbauer polynomials, Bessel integral identities involving up to three Bessel functions have been compiled in integral tables (cf. for instance Refs. [15,16]). Some Bessel function integrals involving up to six Bessel functions can be found in Refs. [1,2]. To proceed one needs analytical expressions for the singular part of the result. However, this problem can be circumvented because in certain kinematical cases (small momentum or specific mass configurations) one can expand the Bessel functions in Taylor series to obtain a series expansion of the Bessel integral. In Ref. [13] we explained how the Bessel integral can be calculated by expanding the Bessel function of the first kind or, in case of vacuum diagrams, one or more McDonald functions. 1 However, in such an expansion the symmetry with respect to the exchanges of the different masses in the diagram is no longer manifest.
In order to avoid this problem, we use another approach to isolate the divergent parts of the integrals. Namely, we use the Gaussian factor e −µ 2 x 2 to protect the integrand against the divergent upper limit. The mass parameter µ in the exponent defines a new regularization scale parameter. In detail, we add and subtract a series expansion of the product of Bessel functions multiplied by e µ 2 x 2 to a specified order. The order of the series expansion is chosen such that the difference is no longer singular at x = 0. The series can be integrated analytically using will be called truncated integral in the following. The truncated integral involves a series truncation With the techniques described in this section we have recalculated the nondegenerate three-loop sunrise-type vacuum diagrams in Refs. [3,4] and have checked our results against the results given in Ref. [3] (diagrams M(1, 1, 1, 1) and U 4 ) and Ref. [4] (diagrams E(u, v, y, z) and F(u, v, y, z)). We have found numerical agreement. This comparison was performed with the help of the code TVID [17] for randomly generated sets of four different masses in a total sample of 1000 mass configurations.

Truncated Bessel integrals for D 0 = 4
For some sunrise-type diagrams, p-space calculations have led to analytic closed form results. These results can be used as input to derive Bessel function integral identities by comparison with the corresponding x-space results. As emphasized before, sunrise-type diagrams in D 0 = 4 dimensions have to be regularized, implying that the corresponding integral identities have to be formulated in terms of truncated Bessel integrals. A complete list of analytical results for vacuum diagrams of sunrise-type topology known up to three-loop order is given in Ref. [4]. We have used the analytical results for E(u, v, y, z) (corresponding to Fig. 1(a)) and F(u, v, y, z) (corresponding to Fig. 1(b)) given in Ref. [4] to calculate truncated Bessel integrals. By choosing µ = 1 we obtain where ℓ a := ln The parameters a and b are mass parameters associated with the vacuum diagrams. All seven of these truncated Bessel function integral identities have been checked numerically using MATHEMATICA.

Bessel integrals and elliptic functions for D 0 = 2
It is well known that the correlator functionΠ(p) for the sunrise-type diagrams is finite in D 0 = 2 dimensions. Therefore, the corresponding Bessel integrals need not to be truncated.
For the vacuum integrals (p 2 = 0) the comparison with results from Refs. [18,19,20,21] results in where λ = λ(m 2 1 , m 2 2 , m 2 3 ) = m 4 1 + m 4 2 + m 4 3 − 2m 2 1 m 2 2 − 2m 2 1 m 2 3 − 2m 2 2 m 2 3 is the Källén function, and Cl p (z) is the Claussen polylogarithm, with arguments α i = 2 arctan( √ −λ/δ i ), where For the more general case p 2 = 0 (cf. Fig. 1(c)), analytical results have been given in Refs. [5,6] in terms of elliptic polylogarithms In order to understand the construction, we start with the complete elliptic integral for the arguments where µ i are the (pseudo)thresholds While the second argument of the elliptic polylogarithm is −1, the last argument is given Defining a modified Claussen dilogarithm bỹ the comparison gives ((i, j, k) is a cyclic permutation of (1, 2, 3)) and the incomplete elliptic integral is given by The result (22)  can be revealed already in the limit p 2 → 0. Obviously, in his limit the main square root simplifies to Further, one has k − = 1 and k + = 0. Thus q → 0, and the elliptic polylogarithms vanish.
However, for the arguments of the (modified) Claussen dilogarithms one obtains leading to Only in the case δ i > 0 (i = 1, 2, 3) and if one is close to the equal mass case, one can reconstruct the α i . As mentioned in Ref. [22], the problem is caused by the standard convention for mathematical software which is in conflict with the Feynman prescription used in quantum field theory. In Ref. [23] the region in the projective plane restricted by λ(m 2 1 , m 2 2 , m 2 3 ) < 0 is called the Källén triangle. In terms of Ref. [23] the restriction δ i > 0 (i = 1, 2, 3) leads to a region enclosed by the three circles of the projective plane about the three corners through the midpoints of the edges, deforming the Källén triangle to a hyperbolic triangle. 2 We emphasize that our configuration space method is free from such kind of problems and, therefore, can be used to check existing analytical results in these problematic cases.

Summary and conclusions
We have recalculated nondegenerate sunrise-type three-loop vacuum integrals in x-space and have found numerical agreement with corresponding results obtained by p-space cal-culations. The agreement of the respective results can be considered to be a necessarybut not sufficient -check on the rather involved p-space three-loop vacuum diagram calculations. It is not difficult to extend the x-space calculation of the sunrise-type vacuum diagrams to higher loop orders where they can again be used to check on the results of higher loop vacuum diagram calculations in p-space when they become available in the future. In fact, the authors of Ref. [25] have checked their p-space results on the degenerate (all masses equal) five-loop sunrise-type vacuum integrals against the x-space results of e.g. Ref. [13].