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Algebraic Algorithms in Perturbative Calculations

Stefan Weinzierl

TL;DR

This work addresses the challenge of computing Feynman loop integrals in perturbation theory by developing algebraic, computer-implementable algorithms based on Hopf algebras. It shows that perturbative expansions can be organized around $Z$-sums and $S$-sums, whose quasi-shuffle Hopf algebra structure provides a systematic path to Laurent expansions in the dimensional-regularization parameter $\varepsilon$, with hypergeometric sums reducible to harmonic polylogarithms. A second Hopf-algebra structure arises for multiple polylogarithms through an integral representation, enabling shuffle-algebra relations and integration-by-parts identities via the antipode. The approach unifies the treatment of loop integrals by converting them into nested sums and polylogarithms, enabling higher-order calculations and robust computer implementations (e.g., FORM, GiNaC).

Abstract

I discuss algorithms for the evaluation of Feynman integrals. These algorithms are based on Hopf algebras and evaluate the Feynman integral to (multiple) polylogarithms.

Algebraic Algorithms in Perturbative Calculations

TL;DR

This work addresses the challenge of computing Feynman loop integrals in perturbation theory by developing algebraic, computer-implementable algorithms based on Hopf algebras. It shows that perturbative expansions can be organized around -sums and -sums, whose quasi-shuffle Hopf algebra structure provides a systematic path to Laurent expansions in the dimensional-regularization parameter , with hypergeometric sums reducible to harmonic polylogarithms. A second Hopf-algebra structure arises for multiple polylogarithms through an integral representation, enabling shuffle-algebra relations and integration-by-parts identities via the antipode. The approach unifies the treatment of loop integrals by converting them into nested sums and polylogarithms, enabling higher-order calculations and robust computer implementations (e.g., FORM, GiNaC).

Abstract

I discuss algorithms for the evaluation of Feynman integrals. These algorithms are based on Hopf algebras and evaluate the Feynman integral to (multiple) polylogarithms.

Paper Structure

This paper contains 8 sections, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Phenomenology
  3. Nested Sums
  4. Expansion of hypergeometric functions
  5. Multiple polylogarithms
  6. The antipode and integration-by-parts
  7. Summary
  8. Notations and conventions

Figures (2)

  • Figure 1: A one-loop Feynman diagram contributing to the process $e^+ e^- \rightarrow q g \bar{q}$.
  • Figure 2: Sketch of the proof for the multiplication of $Z$-sums. The sum over the square is replaced by the sum over the three regions on the r.h.s.