A symbolic summation approach to feynman integrals

Abstract

We discuss new algorithmic strategies for multisums arising in the computation of Feynman parameter integrals, F (N, ε), for N ∈ N, ε ∈ R, and present examples of typical computations coming from the two-loop integrals described in [3] which we encountered during our work on [5]. The genesis of these integrals is described in [4] in detail. Our summation methods have been efficiently implemented in a symbolic toolbox within Mathematica containing F. Stan's package FSums [10, Chapter 3] and C. Schneider's package EvaluateMultiSums, which rely on Sigma [8], J. Ablinger's HarmonicSums [1], and K. Wegschaider's MultiSum [12].The first step of our procedure consists of rewriting Feynman parameter integrals as multisums over hypergeometric terms which fit the input class of our summation algorithms. The sum representations are of the formwhere the summand F (µ, κ.α) is proper hypergeometric in all discrete variables µ i from µ = (µ 1 , . . . , µ p ) and in all summation variables κ j , while the elements of α ∈ C l are additional parameters. For these nested sums, the summation range R ⊆ Z r does not satisfy a finite support condition. In this context, the package FSums takes the sum (1) as input and calls the package MultiSum to obtain a recurrence for its summand, using the WZ-summation strategy [13]. After summing over the algorithmically computed recurrence for the summand F, we determine an inhomogeneous recurrence relation for Sum (µ, α). The inhomogeneous part of this recurrence will contain special instances of the multisum (1) of lower nested depth. Applying the same method on these new sums recursively, we get new recurrences. The recurrences computed at the end of this procedure will have only hypergeometric terms on their right hand sides. For the next step of the method we use procedures from the Sigma package. Namely, these last inhomogeneous difference equations can be viewed in special difference fields introduced by M. Karr [7] and extended significantly by C. Schneider. In this setting, it is possible to find solutions of such recurrences [2,9]. Plugging in these answers into the recurrences from the previous level, 95

Algorithms for special functions

“…Namely, we look for strategies to apply WZ-Fasenmyer summation techniques to large real-world problems of this type. To illustrate our main procedures let us consider the following simple Feynman parameter integral of the type described in [15],…”

“…After introducing the summation range such that we can formally consider all the sums 15) the notion of summability is necessary for the case when the support of F is infinite.…”

“…At the moment, we have analysed and computed all the sums coming from the two-loop integrals described in [11]. These results will appear in the joint publication [15].…”

“…The following 4-fold sum is a typical entry from the list of sums representations for a class of Feynman parameter integrals we computed as part of the work described in [15].…”

“…The work presented in this thesis has lead to four papers [15,38,65,66], three of which are accepted for publication.…”