• Corpus ID: 10885090

Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals

  title={Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals},
  author={Jakob Ablinger and Johannes Bl{\"u}mlein and Mark Round and Carsten Schneider},
Two-point Feynman parameter integrals, with at most one mass and containing local operator insertions in $4+\ep$-dimensional Minkowski space, can be transformed to multi-integrals or multi-sums over hyperexponential and/or hypergeometric functions depending on a discrete parameter $n$. Given such a specific representation, we utilize an enhanced version of the multivariate Almkvist--Zeilberger algorithm (for multi-integrals) and a common summation framework of the holonomic and difference field… 

Extensions of the AZ-algorithm and the Package MultiIntegrate

  • J. Ablinger
  • Mathematics
    Texts & Monographs in Symbolic Computation
  • 2021
The Mathematica package MultiIntegrate, can be considered as an enhanced implementation of the (continuous) multivariate Almkvist Zeilberger algorithm to compute recurrences or differential equations for hyperexponential integrands and integrals.

Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms

The Mellin and inverse Mellin transform is work out which connects the sums under consideration with the associated Poincare iterated integrals, also called generalized harmonic polylogarithms, and algebraic and structural relations are derived for the compactification of S-sum expressions.

Modern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction

A difference field approach for symbolic summation that enables one to simplify such definite nested sums to indefinite nested sums, in particular, in terms of harmonic sums, generalized harmonic sums or binomial sums.

Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations

A new algorithm to solve coupled systems of differential equations in one continuous variable $x$ depending on a small parameter $\epsilon$ if they are expressible in terms of indefinite nested sums and products, based on symbolic summation algorithms in the context of difference rings/fields and uncoupling algorithms is outlined.

Computation of form factors in massless QCD with finite master integrals

We present the bare one-, two-, and three-loop form factors in massless quantum chromodynamics as linear combinations of finite master integrals. Using symbolic integration, we compute their

Simplifying Multiple Sums in Difference Fields

This survey article presents difference field algorithms for symbolic summation that can be solved completely automatically for large scale summation problems for the evaluation of Feynman diagrams in QCD (Quantum ChromoDynamics).

Refined Holonomic Summation Algorithms in Particle Physics

An improved multi-summation approach is introduced and discussed that enables one to simultaneously handle sequences generated by indefinite nested sums and products in the setting of difference

Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

This work uses computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions, which appeared in the famous Handbook of Mathematical Functions but their proofs were lost.



Evaluation of Multi-Sums for Large Scale Problems

This article presents a general summation method based on difference fields that simplifies multi-sums by transforming them from inside to outside to representations in terms of indefinite nested sums and products.

A symbolic summation approach to Feynman integral calculus

Computer Algebra Algorithms for Special Functions in Particle Physics

This work treats iterated integrals of the Poincar\'e and Chen-type, such as harmonic polylogarithms and their generalizations and state an algorithm which rewrites certain types of nested sums into expressions in terms of cyclotomic S-sums.

Algebraic relations between harmonic sums and associated quantities

Algebraic Relations Between Harmonic Sums and Associated Quantities

Nested sums, expansion of transcendental functions and multiscale multiloop integrals

Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools

The Massless Higher-Loop Two-Point Function

We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to

Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials

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

An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities

SummaryIt is shown that every ‘proper-hypergeometric’ multisum/integral identity, orq-identity, with a fixed number of summations and/or integration signs, possesses a short, computer-constructible

Structural relations of harmonic sums and Mellin transforms up to weight w=5