Given a Feynman parameter integral, depending on a single discrete variable $N$ and a real parameter $\epsilon$, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in $\epsilon$. In a first step, the integrals are expressed by hypergeometric multi-sums by means of symbolic transformations. Given this sum format, we develop new summation tools to extract the first coefficients of its series expansion whenever they are expressible in terms of indefinite nested product-sum expressions. In particular, we enhance the known multi-sum algorithms to derive recurrences for sums with complicated boundary conditions, and we present new algorithms to find formal Laurent series solutions of a given recurrence relation.
Starting with single summation over hypergeometric terms developed, e.g., in Gosper (1978); Zeilberger (1990a); Petkovšek (1992); Abramov and Petkovšek (1994); Paule (1995) symbolic summation has been intensively enhanced to multi-summation like, e.g., the holonomic approach of Zeilberger (1990b); Chyzak (2000); Schneider (2005a); Koutschan (2009). In this article we use the techniques of Fasenmyer (1945); Wilf and Zeilberger (1992) which lead to efficient algorithms developed, e.g., in Wegschaider (1997) to compute recurrence relations for hypergeometric multi-sums. Besides this, we rely on multisummation algorithms presented in Schneider (2007) that generalize the summation techniques worked out in Petkovšek et al. (1996); the underlying algorithms are based on a refined difference field theory elaborated in Schneider (2008Schneider ( , 2010) ) that is adapted from Karr's ΠΣ-fields originally introduced in Karr (1981).
We aim at combining these summation approaches which leads to a new framework to evaluate Feynman integrals. In a nutshell, given a Feynman integral, we transform it to hypergeometric multisums, compute afterwards linear recurrences for these multisums, and finally decide constructively by recurrence solving whether the integrals (resp. the multisums) have series expansions whose coefficients can be represented in terms of indefinite nested sums and products. The method consists of a completely algebraic algorithm. It is therefore well-suited for implementation in computer algebra systems.
We show in a first step that Feynman parameter integrals, which contain local operator insertions, in D-dimensional Minkowski space with one time-and (D-1) Euclidean space dimensions, ε = D -4 and ε ∈ R with |ε| ≪ 1, can be transformed by means of symbolic computation to hypergeometric multi-sums S(ε, N ) with N an integer parameter. Given this representation, one can check by analytic arguments whether the integrals can be expanded in a Laurent series w.r.t. the parameter ε, and we seek summation algorithms to compute the first few coefficients of this expansion whenever they are representable in terms of indefinite nested sums and products. Due to the special input class of Feynman integrals, these solutions can be usually transformed to harmonic sums or S-sums; see Blümlein and Kurth (1999); Vermaseren (1999); Moch et al. (2002); Ablinger (2009).
In general, we present an algorithm (see Theorem 1) that decides constructively, if these first coefficients of the ε-expansion can be written in such indefinite nested productsum expressions. Here one first computes a homogeneous recurrence by WZ-theory and Wegschaider’s approach. This recurrence together with initial values gives an alternative representation for the series expansion (see Lemma 1). Moreover, we develop a recurrence solver (see Corollary 1) which computes the coefficients of the expansion in terms of indefinite nested product-sum expressions whenever this is possible. The backbone of this solver relies on algorithms from Petkovšek (1992); Abramov and Petkovšek (1994); Schneider (2001Schneider ( , 2005b)). Since the solutions are highly nested by construction, their simplification to sum representations with minimal depth are crucial; see Schneider (2010).
From the practical point of view there is one crucial drawback of the proposed solution: looking for such recurrences is extremely expensive. For our examples arising from particle physics the proposed algorithm is not applicable considering the available computer and time resources. On that score we relax this very restrictive requirement and search for possibly inhomogeneous recurrence relations. However, the input sums have summands which present poles outside the given summation ranges. Combining Wegschaider’s package MultiSum and the new package FSums presented in Stan (2010) we determine recurrences with inhomogeneous sides consisting of well-defined sums with fewer sum quantifiers. Applying our method to these simpler sums by recursion will lead to an expansion of the right hand side of the starting recurrence. Finally, we compute the coefficients of the original input sum by our new recurrence solver mentioned above.
The outline of the article is as follows. In Section 2 we explain all computation steps that lead from Feynman integrals to hypergeometric multi-sums of the form (7) which can be expanded in a Laurent expansion (11) where the coefficients F i (N ) can be represented in the form (12). In the beginning of Section 3 we face the problem that the multisums (7) have to be split further in the form (13) to fit the input class of our summation algorithms. We first discuss convergent sums only. The treatment of those sums which diverge in this special format or sums with several infinite summations that have difficult convergence properties will be dealt with later, cf. Remark 5. In the remaining parts of Section 3 we present the general mechanisms to compute the first coeff
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