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- XSummer - Transcendental Functions and Symbolic Summation in Form

S. Moch, P. Uwer

TL;DR

The paper presents XSummer, a Form-based package for symbolic summation of nested sums (S-sums) that arise in higher-order perturbative quantum field theory calculations. It introduces the mathematical foundation of S-sums and their relation to Z-sums and multiple polylogarithms, and then describes four recursive algorithm types (A–D) that drive the algebraic manipulation and reduction of these sums. XSummer provides user-facing interfaces and a suite of internal routines to perform multi-scale summations, including harmonic sums in infinity via tables linked to Summer’s MZV data. The authors demonstrate extensive examples and discuss performance considerations, highlighting the package’s utility for expanding generalized hypergeometric functions around small parameters and producing closed-form results in terms of polylogarithms, with implications for high-precision QCD calculations and broader symbolic-sum applications.

Abstract

Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their generalizations appear as building blocks, originating for example from the expansion of generalized hypergeometric functions around integer values of the parameters. In this Letter we discuss the implementation of several algorithms to solve these sums by algebraic means, using the computer algebra system Form.

- XSummer - Transcendental Functions and Symbolic Summation in Form

TL;DR

The paper presents XSummer, a Form-based package for symbolic summation of nested sums (S-sums) that arise in higher-order perturbative quantum field theory calculations. It introduces the mathematical foundation of S-sums and their relation to Z-sums and multiple polylogarithms, and then describes four recursive algorithm types (A–D) that drive the algebraic manipulation and reduction of these sums. XSummer provides user-facing interfaces and a suite of internal routines to perform multi-scale summations, including harmonic sums in infinity via tables linked to Summer’s MZV data. The authors demonstrate extensive examples and discuss performance considerations, highlighting the package’s utility for expanding generalized hypergeometric functions around small parameters and producing closed-form results in terms of polylogarithms, with implications for high-precision QCD calculations and broader symbolic-sum applications.

Abstract

Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their generalizations appear as building blocks, originating for example from the expansion of generalized hypergeometric functions around integer values of the parameters. In this Letter we discuss the implementation of several algorithms to solve these sums by algebraic means, using the computer algebra system Form.

Paper Structure

This paper contains 11 sections, 22 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Harmonic sums and their generalizations
  3. Algorithms
  4. The XSummer package
  5. Basic syntax --- form follows function
  6. List of procedures provided by XSummer
  7. Procedures to be called directly
  8. Internal procedures
  9. Harmonic sums in infinity
  10. Examples
  11. Conclusion

Figures (1)

  • Figure 3.1: Internal structure of the XSummer package