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Harmonic Sums, Polylogarithms, Special Numbers, and their Generalizations

Jakob Ablinger, Johannes Blümlein

TL;DR

Massless and massive multi-loop Feynman calculations generate a hierarchy of nested sums, iterated integrals, and special constants that are organized by shuffle and stuffle algebras and represented via Mellin transforms. The paper surveys harmonic sums, harmonic polylogarithms, and their generalizations (S-sums, cyclotomic sums, binomial-weighted sums), clarifying their algebraic/structural relations and analytic continuation to complex $N$. It links these objects to multiple zeta values and discusses basis counting and computational tools (HarmonicSums) that enable efficient, high-precision perturbative calculations. The framework provides a unified language for high-loop quantum field theory, guiding both analytic and numerical evaluation of observables across increasingly complex kinematic configurations.

Abstract

In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops. These quantities are elements of stuffle and shuffle algebras implying algebraic relations being widely independent of the special quantities considered. They are supplemented by structural relations. The generalizations are given in terms of generalized harmonic sums, (generalized) cyclotomic sums, and sums containing in addition binomial and inverse-binomial weights. To all these quantities iterated integrals and special numbers are associated. We also discuss the analytic continuation of nested sums of different kind to complex values of the external summation bound N.

Harmonic Sums, Polylogarithms, Special Numbers, and their Generalizations

TL;DR

Massless and massive multi-loop Feynman calculations generate a hierarchy of nested sums, iterated integrals, and special constants that are organized by shuffle and stuffle algebras and represented via Mellin transforms. The paper surveys harmonic sums, harmonic polylogarithms, and their generalizations (S-sums, cyclotomic sums, binomial-weighted sums), clarifying their algebraic/structural relations and analytic continuation to complex . It links these objects to multiple zeta values and discusses basis counting and computational tools (HarmonicSums) that enable efficient, high-precision perturbative calculations. The framework provides a unified language for high-loop quantum field theory, guiding both analytic and numerical evaluation of observables across increasingly complex kinematic configurations.

Abstract

In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops. These quantities are elements of stuffle and shuffle algebras implying algebraic relations being widely independent of the special quantities considered. They are supplemented by structural relations. The generalizations are given in terms of generalized harmonic sums, (generalized) cyclotomic sums, and sums containing in addition binomial and inverse-binomial weights. To all these quantities iterated integrals and special numbers are associated. We also discuss the analytic continuation of nested sums of different kind to complex values of the external summation bound N.

Paper Structure

This paper contains 17 sections, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Polylogarithms, Nielsen Integrals, Harmonic Polylogarithms
  3. Harmonic Sums
  4. Algebraic and Structural Relations
  5. Algebraic Relations
  6. Structural Relations
  7. Harmonic Polylogarithms
  8. Harmonic Sums
  9. Multiple Zeta Values
  10. Generalized Harmonic Sums and Polylogarithms
  11. Generalized Harmonic Sums
  12. Generalized Harmonic Polylogarithms
  13. Relations between $S$-Sums at Infinity
  14. Cyclotomic Harmonic Sums and Polylogarithms and their Generalization
  15. Nested Binomial and Inverse-binomial Harmonic Sums and Associated Polylogarithms
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Relations between the different extensions of harmonic sums.
  • Figure 3: Connection between harmonic sums (H-Sums), S-sums (S-Sums) and cyclotomic harmonic sums (C-Sums), their values at infinity and harmonic polylogarithms (H-Logs), generalized harmonic polylogarithms (G-Logs) and cyclotomic harmonic polylogarithms (C-Logs) and their values at special constants.