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A Symbolic Summation Approach to Feynman Integral Calculus
"... Given a Feynman parameter integral, depending on a single discrete variable N and a real parameter ε, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in ε. In a first step, the integrals are expressed by hypergeometric multisums by means of sy ..."
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Given a Feynman parameter integral, depending on a single discrete variable N and a real parameter ε, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in ε. In a first step, the integrals are expressed by hypergeometric multisums 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 productsum expressions. In particular, we enhance the known multisum 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.
arXiv:0902.4091 [hepph]
, 902
"... Determining the closed forms of the O(a3 s) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra ..."
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Determining the closed forms of the O(a3 s) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra
Determining the closed forms of the O(a³ s) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra
, 2009
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arXiv:1105.6063 [mathph] Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
, 2011
"... The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré– iterated integrals including denomina ..."
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The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré– iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, resp., for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight w = 1,2 sums up to cyclotomy l = 20.
Nikhef Theory Group
, 907
"... We provide a data mine of proven results for multiple zeta values (MZVs) of the form ζ(s1,s2,...,sk) = ∑ ∞ { s1 n1>n2>...>nk>0 1/(n1...nsk k)} with weight w = ∑ k i=1 si and depth k and for Euler sums of the form ∑ ∞ { n1 n1>n2>...>nk>0 (ε1...εnk 1)/(ns1 1...nsk k)} with signs εi = ±1. Notably, ..."
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We provide a data mine of proven results for multiple zeta values (MZVs) of the form ζ(s1,s2,...,sk) = ∑ ∞ { s1 n1>n2>...>nk>0 1/(n1...nsk k)} with weight w = ∑ k i=1 si and depth k and for Euler sums of the form ∑ ∞ { n1 n1>n2>...>nk>0 (ε1...εnk 1)/(ns1 1...nsk k)} with signs εi = ±1. Notably, we achieve explicit proven reductions of all MZVs with weights w ≤ 22, and all Euler sums with weights w ≤ 12, to bases whose dimensions, bigraded by weight and depth, have sizes in precise agreement with the Broadhurst–
Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
, 2013
"... In recent three–loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the socalled generalized harmonic sums (in short Ssums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In th ..."
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In recent three–loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the socalled generalized harmonic sums (in short Ssums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincaré iterated integrals, also called generalized harmonic polylogarithms. In this regard, we obtain explicit analytic continuations by means of asymptotic expansions of the Ssums which started to occur frequently in current QCD calculations. In addition, we derive algebraic and structural relations, like differentiation w.r.t. the external summation index and different multiargument relations, for the compactification of Ssum expressions. Finally, we calculate algebraic relations for infinite Ssums, or equivalently for generalized harmonic polylogarithms evaluated at special values. The corresponding algorithms and relations are encoded in the computer algebra package HarmonicSums. 1