Evaluation of Multi-Sums for Large Scale Problems

Abstract

A big class of Feynman integrals, in particular, the coefficients of their Laurent series expansion w.r.t.\ the dimension parameter $\ep$ can be transformed to multi-sums over hypergeometric terms and harmonic sums. In this article, we present a general summation method based on difference fields that simplifies these multi--sums by transforming them from inside to outside to representations in terms of indefinite nested sums and products. In particular, we present techniques that assist in the task to simplify huge expressions of such multi-sums in a completely automatic fashion. The ideas are illustrated on new calculations coming from 3-loop topologies of gluonic massive operator matrix elements containing two fermion lines, which contribute to the transition matrix elements in the variable flavor scheme.

9 pages Latex, Contr. Proc. RADCOR 2011

References in corpus (11)

• Parton distribution functions and benchmark cross sections at NNLO
• Mellin Moments of the {$O({\alpha }_s^3$)} Heavy Flavor Contributions to unpolarized Deep-Inelastic Scattering at $Q^2\gg m^2$ and Anomalous Dimensions
• Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5
• A Refined Difference Field Theory for Symbolic Summation
• The Gluonic Operator Matrix Elements at O(α_s^2) for DIS Heavy Flavor Production
• Two--Loop Massive Operator Matrix Elements for Unpolarized Heavy Flavor Production to $O(ε)
• A Symbolic Summation Approach to Feynman Integral Calculus
• $O({\alpha }_s^2)$ and $O({\alpha }_s^3)$ Heavy Flavor Contributions to Transversity at $Q^2\gg m^2$
• 3-Loop Heavy Flavor Corrections to DIS with two Massive Fermion Lines
• Parameterized Telescoping Proves Algebraic Independence of Sums
• New Heavy Flavor Contributions to the DIS Structure Function $F_2(x,Q^2)$ at $O(α_s^3)

Cited by in corpus (30)

• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• The 3-Loop Pure Singlet Heavy Flavor Contributions to the Structure Function $F_2(x,Q^2)$ and the Anomalous Dimension
• Computer Algebra Algorithms for Special Functions in Particle Physics
• Iterated Binomial Sums and their Associated Iterated Integrals
• Modern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction
• Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms
• Massive 3-loop Ladder Diagrams for Quarkonic Local Operator Matrix Elements
• The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements
• The $O({\alpha }_s^3{n}_f{T}_F^2{C}_{A,F})$} Contributions to the Gluonic Massive Operator Matrix Elements
• The Three-Loop Splitting Functions $P_{qg}^{(2)}$ and $P_{gg}^{(2,N_F)}$
• The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
• The Heavy Quark Form Factors at Two Loops
• The Heavy Fermion Contributions to the Massive Three Loop Form Factors
• On the relation between the $\overline{\mathrm{MS}}$ and the kinetic mass of heavy quarks
• The three-loop single mass polarized pure singlet operator matrix element
• The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$
• Subleading Logarithmic QED Initial State Corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }/{Z^0}^{\ast }$ to $O({\alpha }^6{L}^5)$
• The three-loop polarized pure singlet operator matrix element with two different masses
• Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals
• Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations
• Higher Order Heavy Quark Corrections to Deep-Inelastic Scattering
• New Results on the 3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering
• Three-Loop Contributions to the Gluonic Massive Operator Matrix Elements at General Values of N
• General $\epsilon$-representation for scalar one-loop Feynman integrals
• 3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering with Two Heavy Quark Lines
• Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams
• 3-Loop Corrections to the Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering
• 3-loop Massive $O(T_F^2)$ Contributions to the DIS Operator Matrix Element $A_{gg}$
• Recent progress on the calculation of three-loop heavy flavor Wilson coefficients in deep-inelastic scattering
• 3-loop heavy flavor Wilson coefficients in deep-inelastic scattering