A Symbolic Summation Approach to Feynman Integral Calculus

Abstract

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.

References in corpus (11)

• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• 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
• A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics
• A Refined Difference Field Theory for Symbolic Summation
• Two-Loop Massive Operator Matrix Elements and Unpolarized Heavy Flavor Production at Asymptotic Values Q^2 >> m^2
• 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)
• The Longitudinal Heavy Quark Structure Function F_{L}^{Q\bar{Q}} in the Region Q^2 \gg m^2 at O(α_s^3)
• Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams
• A Symbolic Summation Approach to Find Optimal Nested Sum Representations
• Difference Equations in Massive Higher Order Calculations

Cited by in corpus (26)

• 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
• Computer Algebra Algorithms for Special Functions in Particle Physics
• 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
• HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: Horn hypergeometric functions of two variables
• HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions pFq, F1,F2,F3,F4
• The O(α_s^3 T_F^2) Contributions to the Gluonic Operator Matrix Element
• Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions
• The Method of Arbitrarily Large Moments to Calculate Single Scale Processes in Quantum Field Theory
• 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 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
• 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
• Algorithms to Evaluate Multiple Sums for Loop Computations
• A toolbox to solve coupled systems of differential and difference equations
• A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams
• Three-Loop Contributions to the Gluonic Massive Operator Matrix Elements at General Values of N
• Higher Order Heavy Quark Corrections to Deep-Inelastic Scattering
• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• Integral representations of equally positive integer-indexed harmonic sums at infinity