A Refined Difference Field Theory for Symbolic Summation

Abstract

In this article we present a refined summation theory based on Karr's difference field approach. The resulting algorithms find sum representations with optimal nested depth. For instance, the algorithms have been applied successively to evaluate Feynman integrals from Perturbative Quantum Field Theory.

Uses elseart.cls and yjsco.sty

References in corpus (6)

• Harmonic sums, Mellin transforms and Integrals
• The third-order QCD corrections to deep-inelastic scattering by photon exchange
• Harmonic Sums and Mellin Transforms up to two-loop Order
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• Gaussian hypergeometric series and supercongruences
• Difference Equations in Massive Higher Order Calculations

Cited by in corpus (55)

• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra
• The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity
• 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
• The Theory of Deeply Inelastic Scattering
• The O(α_s^3) Massive Operator Matrix Elements of O(n_f) for the Structure Function F_2(x,Q^2) and Transversity
• Two--Loop Massive Operator Matrix Elements for Unpolarized Heavy Flavor Production to $O(ε)
• Iterated Binomial Sums and their Associated Iterated Integrals
• The Transition Matrix Element A_{gq}(N) of the Variable Flavor Number Scheme at O(α_s^3)
• Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable
• 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
• Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams
• Massive 3-loop Ladder Diagrams for Quarkonic Local Operator Matrix Elements
• Determining the closed forms of the $O(a_s^3)$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra
• The $O({\alpha }_s^3{n}_f{T}_F^2{C}_{A,F})$} Contributions to the Gluonic Massive Operator Matrix Elements
• A Symbolic Summation Approach to Feynman Integral Calculus
• The Three-Loop Splitting Functions $P_{qg}^{(2)}$ and $P_{gg}^{(2,N_F)}$
• The O(α_s^3 T_F^2) Contributions to the Gluonic Operator Matrix Element
• The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
• The Method of Arbitrarily Large Moments to Calculate Single Scale Processes in Quantum Field Theory
• Generalized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers
• A Symbolic Summation Approach to Find Optimal Nested Sum Representations
• The Heavy Fermion Contributions to the Massive Three Loop Form Factors
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• Evaluation of Multi-Sums for Large Scale Problems
• The Polarized Transition Matrix Element $A_{gq}(N)$ of the Variable Flavor Number Scheme at $O({\alpha }_s^3)$
• 3-Loop Heavy Flavor Corrections to DIS with two Massive Fermion Lines
• The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$
• The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg,Q}$ and $\Delta A_{gg,Q}$
• Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals
• The Logarithmic Contributions to the Polarized and Operator Matrix Elements in Deeply Inelastic Scattering
• New Heavy Flavor Contributions to the DIS Structure Function $F_2(x,Q^2)$ at $O(α_s^3)
• A Difference Ring Theory for Symbolic Summation
• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals
• Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations
• The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms
• Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums
• Logarithmic $O({\alpha }_s^3)$ contributions to the DIS Heavy Flavor Wilson Coefficients at $Q^2\gg m^2$
• Algorithms to Evaluate Multiple Sums for Loop Computations
• Representing ($q$--)hypergeometric products and mixed versions in difference rings
• Heavy Flavor DIS Wilson coefficients in the asymptotic regime
• A toolbox to solve coupled systems of differential and difference equations
• Three-Loop Contributions to the Gluonic Massive Operator Matrix Elements at General Values of N
• Higher Order Heavy Quark Corrections to Deep-Inelastic Scattering
• New Results on the 3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering
• A case study for $\zeta (4)$
• 3-Loop Corrections to the Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering
• 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 Massive $O(T_F^2)$ Contributions to the DIS Operator Matrix Element $A_{gg}$
• The QCD Coupling and Parton Distributions at High Precision
• Recent progress on the calculation of three-loop heavy flavor Wilson coefficients in deep-inelastic scattering