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A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics

Jakob Ablinger

arXiv:1011.1176v1math-ph

TL;DR

The paper addresses symbolic manipulation of harmonic sums and related quantities needed for compact analytic calculations. It develops HarmonicSums with algebraic, transform-based, and rewriting methods, and illustrates their use with Sigma for Feynman-integral evaluation.

  • Problem

    Relations among finite harmonic sums are needed to simplify calculations and obtain compact analytic results, motivating a unified treatment of harmonic sums and related objects.

  • Method

    The paper develops HarmonicSums using quasi-shuffle algebra, algebraic-independence methods, extended-Mellin differentiation, and an algorithm for rewriting certain nested sums.

  • Results

    The paper shows that harmonic sums form a quasi-shuffle algebra, connects them with Lyndon words, and rewrites certain nested sums in terms of harmonic sums.

  • Takeaways & Limitations

    The algorithms, used with Sigma, support the evaluation of Feynman integrals through symbolic manipulation of harmonic sums and related expressions.

  • Takeaways & Limitations

    The rewriting approach does not cover all sums, and relation (3.3) does not hold when one of the objects is divergent.

Abstract

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In this work we present the computer algebra package HarmonicSums and its theoretical background for the manipulation of harmonic sums and some related quantities as for example Euler-Zagier sums and harmonic polylogarithms. Harmonic sums and generalized harmonic sums emerge as special cases of so-called d'Alembertian solutions of recurrence relations. We show that harmonic sums form a quasi-shuffle algebra and describe a method how we can find algebraically independent harmonic sums. In addition, we define a differentiation on harmonic sums via an extended version of the Mellin transform. Along with that, new relations between harmonic sums will arise. Furthermore, we present an algorithm which rewrites certain types of nested sums into expressions in terms of harmonic sums. We illustrate by nontrivial examples how these algorithms in cooperation with the summation package Sigma support the evaluation of Feynman integrals.

Eidesstattliche Erkl¨arung

The paper presents HarmonicSums and its theoretical framework for manipulating harmonic sums and related objects, with algorithms supporting nested-sum rewriting and Feynman-integral calculations.

  • HarmonicSums is a computer algebra package for processing harmonic sums, Euler-Zagier sums, and harmonic polylogarithms.
  • Harmonic and generalized harmonic sums arise as special cases of d’Alembertian solutions of recurrence relations.
  • Harmonic sums form a quasi-shuffle algebra, and the paper presents a method for finding algebraically independent sums.
  • An extended Mellin transform defines differentiation on harmonic sums and yields new relations between them.
  • An algorithm rewrites certain nested sums in terms of harmonic sums, with examples showing cooperation with Sigma in evaluating Feynman integrals.

4 Half-Integer Relations

This passage defines notation for harmonic sums, Euler-Zagier sums, harmonic polylogarithms, transforms, algebraic structures, and word operations used in the paper.

  • S_a1,a2,...(n) denotes a harmonic sum, while Z_a1,a2,...(n) denotes an Euler-Zagier sum.
  • H_m1,m2,...(x) denotes a harmonic polylogarithm, and M(f(x),n) and M+(f(x),n) denote the Mellin and extended Mellin transforms.
  • S_p(n) and S(n) denote polynomial sets in harmonic sums, while I_p and I denote corresponding ideals.
  • The shuffle product and quasi-shuffle product are denoted by their respective product operations.
  • A* is the free monoid over A, A+ contains non-empty words, and Lyndon(A) contains Lyndon words.
  • The notation also defines sign, word degree, absolute value, Möbius function, and all-one and all-zero vectors.

Introduction

Harmonic sums and related structures support compact analytic work in particle physics, where analytic continuation, structural relations, and transformations are needed. HarmonicSums combines algebraic and analytic procedures with nested-sum algorithms, while Sigma helps simplify sums and evaluate Feynman integrals.

  • Motivation: Harmonic sums and related multiple sums provide a convenient description of field-theoretic quantities and help obtain compact analytic results.The introduction connects these sums to particle-physics calculations and emphasizes simplifying finite harmonic-sum expressions.
  • Motivation: Physical applications require analytic continuation of nested harmonic sums and eventually their complex analysis.The discussion considers extensions from integer indices toward rational and real values.
  • Relations: Allowing rational indices yields half-integer relations, while real indices permit differentiation with respect to n and new structural relations.These extensions broaden the relations available for harmonic sums beyond their finite-index algebra.
  • HarmonicSums: HarmonicSums provides operations for harmonic sums, harmonic polylogarithms, Euler-Zagier sums, and S-sums, including Mellin transforms, differentiation, and relation handling.The package also includes procedures and tables for algebraic and structural relations.
  • Applications: Algorithms rewrite certain nested sums in terms of harmonic sums, while Sigma simplifies nested depth and denominator degree before supporting Feynman-integral evaluation.The paper illustrates this cooperation through a nontrivial example.
  • Algebraic structure: Harmonic sums form a quasi-shuffle algebra connected to Lyndon words, supporting the study and computation of algebraic relations.The paper links this algebraic structure to algebraically independent sums and procedures for finding and applying relations.

Algebraic Relations between Multiple Harmonic Sums

The section develops algebraic structures for multiple harmonic sums, showing that their products reduce to linear combinations and that quasi-shuffle algebras admit Lyndon-word polynomial representations. It also relates harmonic sums to Euler-Zagier sums and provides package procedures for these transformations.

  • Algebraic independence: Lyndon words provide a route to determining the number of algebraically independent harmonic sums in the quasi-shuffle setting.The resulting count is also an upper bound when harmonic sums are considered as sequences.
  • Harmonic-sum notation: Harmonic sums are defined by depth and weight, with weight given by the sum of the absolute values of their indices.The modified notation restricts indices to 1, 0, and −1 while encoding larger absolute indices through zero markers.
  • Computer-algebra procedures: The package automates multiplication and expansion of harmonic and Euler-Zagier sums, producing explicit linear combinations of nested sums.Examples include products involving signed indices and expanded Z-sum expressions.
  • Euler-Zagier sums: Multiple harmonic sums can be converted recursively to Euler-Zagier sums, and the HarmonicSums package provides procedures for both conversion directions.The package examples show explicit S- and Z-sum outputs for these transformations.
  • Quasi-shuffle algebra: The quasi-shuffle algebra is a free polynomial algebra on elements associated with Lyndon words.This establishes a polynomial generating structure for the algebra.
  • Harmonic-sum products: Harmonic sums form a quasi-shuffle algebra, allowing products with a common upper limit to be rewritten as linear combinations of single harmonic sums.This multiplication follows from the quasi-shuffle product and is illustrated by explicit nested-sum expansions.

2.5 The Number of Lyndon Words or Basic Sums

The section counts algebraically independent harmonic sums through Lyndon-word methods and develops relations that reduce nested sums to basic sums. For weight 3, the strategy reaches the maximal number of independent sums and solves the problem completely.

  • Counting basic sums: The number of algebraically independent sums is counted for fixed index sets using formulas related to Lyndon words and Witt formulas.The paper distinguishes sums in S(n)/I, sums viewed as sequences, and basic sums represented in Table 2.1.
  • Counting basic sums: The paper verifies the figures in Table 2.1 up to weight 7, but treats them as upper bounds without a rigorous proof that the relevant quotient spaces coincide.The qualification applies when comparing S(n)/I with S(n)/∼ as sequences.
  • Deriving relations: Shuffle and quasi-shuffle relations are used to express harmonic sums of a given depth in terms of a minimal set of algebraically independent sums.The construction determines algebraic relations up to weight 7 and uses explicit systems of equations for different index patterns.
  • Deriving relations: For depth-three sums with three different indices, a rank-4 system expresses the six permutations through two chosen depth-three sums and lower-depth sums.Analogous systems handle two different indices and three equal indices.
  • Results and implementation: Applying the second Witt formula to depth-three sums finds the maximal number of algebraically independent sums and solves weight 3 completely.HarmonicSums computes these relations, with relations up to weight 6 included in the package.

Harmonic Polylogarithms and the Differentiation of Harmonic Sums

Harmonic polylogarithms are connected to multiple harmonic sums through Mellin and inverse Mellin transforms, enabling analytic continuation and a differentiation framework. HarmonicSums implements and verifies these procedures, and the resulting relations reduce algebraic bases.

  • Mellin-transform connection: Harmonic polylogarithms and multiple harmonic sums are connected through a special extension of the Mellin transform.Mellin transforms yield harmonic-sum expressions, while inverse Mellin transforms provide harmonic-polylogarithm connections.
  • Mellin-transform connection: Inverse Mellin transforms can construct analytic continuations of multiple harmonic sums.
  • Algorithms and verification: The chapter analyzes and verifies algorithms for Mellin and inverse Mellin transforms, including endpoint quantities at infinity and one.
  • Computer-algebra implementation: HarmonicSums implements Mellin-transform, inverse-Mellin, harmonic-polylogarithm, and harmonic-sum differentiation procedures.
  • Computer-algebra implementation: Differentiation generates new relations between multiple harmonic sums and reduces the computed algebraic basis.

3.1 Definition of Harmonic Polylogarithms

The section defines harmonic polylogarithms inductively, develops their shuffle algebra, and provides procedures for handling products, endpoint singularities, and logarithmic factors. It shows that divergences at x →1 arise from leading ones and can be isolated systematically.

  • Definitions: Harmonic polylogarithms are C∞((0, 1)) functions defined inductively using indices from {0, 1, −1}.
  • Shuffle products: Products of harmonic polylogarithms can be rewritten as sums of single harmonic polylogarithms through the shuffle product.
  • Computer-algebra procedures: HarmonicSums provides procedures for carrying out harmonic-polylogarithm products and extracting logarithmic factors.
  • Endpoint behavior: Trailing zeros permit decomposition into a polynomial in H0(x), with highest power equal to the number of trailing zeros.
  • Endpoint behavior: Leading ones permit decomposition into a polynomial in H1(x), with highest power equal to their count.
  • Endpoint behavior: All divergences as x →1 can be traced to H1(x) = −log(1 −x), while removing leading ones and trailing zeros yields continuous coefficients.

3.2 Multiple Harmonic Sums at Infinity

Multiple harmonic sums at infinity are finite exactly under an index condition, and their divergent behavior can be isolated in the basic sum S1(n). This supports polynomial decompositions and evaluation through convergent sums and zeta-values.

  • Convergence at infinity: A multiple harmonic sum Sa1,a2,...,ap(n) converges as n →∞ if and only if a1 ≠ 1.
  • Decomposition: Every multiple harmonic sum can be expressed using sums without leading ones and sums of the form S1w(n).
  • Decomposition: Sums of the form S1w(n) can be reduced to sums Si(n), enabling a polynomial representation in S1(n) with convergent harmonic-sum coefficients.
  • Divergences: All divergences of multiple harmonic sums can be traced back to S1(n).
  • Values at infinity: For k > 1, harmonic sums Sk(∞) become zeta-values.

3.3 Values at One and Power Series Expansion of Harmonic Polylogarithms

The section relates harmonic-polylogarithm values at x = 1 and power-series expansions to harmonic sums at infinity. It handles endpoint divergences by extracting logarithmic factors and uses finite-cutoff shuffle algebra to define products consistently.

  • Values at one: At x = 1, trailing-zero factors vanish through powers of log(x), leaving constant terms when coefficients are continuous.
  • Values at one: Harmonic-polylogarithm limits at x = 1 correspond to harmonic sums at infinity, with leading ones producing divergent values.
  • Computer-algebra evaluation: HarmonicSums evaluates harmonic polylogarithms at one and rewrites the results as harmonic sums using TransformToSSums.
  • Divergent endpoint products: The product formula for harmonic polylogarithms fails when an object is divergent, so finite-cutoff sums are used to cancel divergences before taking the limit.
  • Divergent endpoint products: This finite-cutoff procedure provides a consistent basis for defining the Mellin transform.

3.4 The Mellin-Transform of Harmonic Polylogarithms

The section defines an extended Mellin transform for harmonic polylogarithms and develops recursive procedures for computing it. Endpoint singularities are isolated or handled formally so divergent terms cancel in the final expressions.

  • Definition: The Mellin transform is defined for locally integrable functions on (0,1) when the integral converges.The extended construction is subsequently denoted simply by M.
  • Endpoint behavior: Harmonic polylogarithms with leading ones can be singular at x = 1, so powers of log(1 −x) are extracted before evaluating the remaining functions there.After extraction, the remaining harmonic polylogarithms are finite at x = 1.
  • Convergence: Convergence is established by reducing the relevant integrals to bounded harmonic polylogarithms multiplied by powers of logarithms.The proof uses continuity and convergence of the resulting integrals on the endpoint interval.
  • Formal treatment: The recursion may use formal equalities when individual sides diverge, because divergent parts cancel and the final Mellin-transform expressions are finite.The analytic proof that the formal recursions equal the Mellin transforms is omitted.

3.5 The Inverse Mellin Transform of Multiple Harmonic Sums

The inverse Mellin transform represents multiple harmonic sums through functions and distributions, while shuffle-product structure organizes the resulting expressions. A complexity ordering identifies a unique leading harmonic sum in weighted harmonic-polylogarithm transforms.

  • Shuffle structure: The shuffle product is organized by mergings that preserve the relative order of the entries in both index vectors.This structure supports expansions of products into multiple harmonic sums.
  • Complexity ordering: A complexity order compares harmonic sums by weight and depth, although it is not a linear ordering.Consequently, a set can contain more than one most complicated sum.
  • Leading sums: For Mellin transforms weighted by 1/(1 −x) or 1/(1 + x), exactly one most complicated harmonic sum appears, while the remaining sums are smaller and linear.The maximal weight is at most the harmonic polylogarithm weight plus one.
  • Algorithm: Algorithm 2 computes the most complicated harmonic sum by enumerating recursion branches and reducing their products through shuffle relations.The construction yields an injective mapping from weighted polylogarithms to their most complicated harmonic sums.

3.6 Differentiation of Multiple Harmonic Sums

The paper defines differentiation of harmonic sums through inverse and forward Mellin transforms after multiplying by H0(x). These differential relations generate additional reductions and can shrink algebraic bases.

  • Analytic continuation: The Mellin-transform representation provides analytic continuation of harmonic sums to n ∈R, enabling differentiation with respect to n.The differentiated result can then be transformed back into harmonic sums.
  • Differentiation procedure: Differentiation with respect to n is implemented by inverse-transforming a harmonic sum, removing constant terms, multiplying by H0(x), and applying the Mellin transform.The package provides this procedure as an algorithm for differentiating harmonic sums.
  • New relations: Differentiation introduces new relations among multiple harmonic sums that can express some sums through less complicated ones.For example, depth-one sums can be represented using derivatives of S±1(n).
  • Limitation: The method requires the actual indices of a harmonic sum, so the alternative strategy cannot be adopted when those indices are unavailable.This is identified as a limitation of the differentiation-based reduction strategy.
  • Basis reduction: Using algebraic and differential relations, the original basis can be reduced to algebraically independent sums with as many sums eliminated by differentiation as possible.Table 3.1 compares the numbers of basic sums obtained with algebraic relations alone and with differential relations added.

Half-Integer Relations

The paper extends harmonic-sum reductions by considering half-integer arguments and derives additional relations from this broader domain. Combining algebraic, differential, and half-integer relations reduces the required bases further.

  • Basis reduction: The number of basic sums decreases when algebraic relations are combined with half-integer relations, compared with algebraic relations alone.Tables 4.1 and 4.2 report these reductions for the considered sum families.
  • Combined relations: Combining algebraic, differential, and half-integer relations reduces the number of basic sums further.The additional reduction is shown in Tables 4.1 and 4.2.
  • Depth-one example: For depth-one sums, differentiation and half-integer relations express all sums using S1(n).In particular, S−1(n) can be represented using S1(n) and S1(n/2).
  • Algebraic independence: The tabulated d-, h-, and dh-basic sums are algebraically independent by construction.The package provides tables using algebraic and half-integer relations up to weight 6.

Summation of Multiple Harmonic Sums

This section develops formulas and an algorithm for rewriting selected nested sums into harmonic-sum expressions. The approach combines quasi-shuffle relations, recursive identities, partial fractions, and Sigma-assisted summation.

  • Implementation: The resulting implementation is available through TransformToSSums, while Sigma handles complementary nested sums and HarmonicSums seeks closed forms.The final expressions can include rational functions in n, harmonic sums, (−1)^n, and residual sums over the remaining denominator.
  • Algebraic reduction: Quasi-shuffle algebra splits products and nested expressions into combinations of multiple harmonic sums.This supplies algebraic reductions before applying summation identities.
  • Problem: The target problem is to represent sums of the specified forms as far as possible in terms of harmonic sums.The sums may involve rational functions, polynomial factors, alternating factors, and shifted harmonic sums.
  • Polynomial factors: Polynomial factors are handled recursively, yielding combinations of harmonic sums, rational functions in n, and possibly (−1)^n.The method covers arbitrary depth and powers of the summation index.
  • Shift reduction: Recursive shift identities reduce sums involving S_a(n±c) to sums with upper index n and reduced depth.These identities are applied repeatedly to express shifted sums in the standard harmonic-sum basis.
  • Rational factors: For general rational functions, the denominator is factored and split into structured factors before partial-fraction and recursive shift manipulations.The decomposition separates factors of the form (i+c)^m from the remaining denominator.

4. Split the sum into two sums, each sum over one fraction, i.e.,

The section demonstrates how Sigma and HarmonicSums cooperate to simplify nested sums. The workflow produces closed forms when possible, while explicitly recognizing sums that cannot be rewritten entirely in harmonic sums.

  • Decomposition: Partial-fraction decomposition separates the summand into fractions that can be treated by the harmonic-sum theorems.The implementation may use coefficient solving rather than the extended Euclidean algorithm.
  • Sigma integration: Sigma simplifies residual nested sums before passing the result back to HarmonicSums for harmonic-sum conversion.The resulting denominator is reduced to minimal degree when possible.
  • Closed forms: HarmonicSums finds a closed form in terms of harmonic sums when the transformed sum admits one.The output may contain rational functions and nested harmonic sums in the upper index.
  • Examples: The examples include both comparatively simple sums and more difficult nested sums handled by the same transformation framework.These examples illustrate outputs involving several nested harmonic-sum terms.
  • Scope: Not every sum can be rewritten in terms of harmonic sums, leaving a scope boundary for the method.The section explicitly presents an example outside the fully reducible class.

An Example from Particle Physics

The particle-physics example applies HarmonicSums and Sigma to a substantial expression containing many harmonic sums. Algebraic, differential, and half-integer relations progressively reduce its representation while preserving a closed-form evaluation.

  • Application context: The example concerns single-scale quantities in particle physics, including anomalous dimensions and hard-scattering cross sections.The surrounding discussion connects these quantities with finite-order difference equations in Mellin space.
  • Closed-form evaluation: Sigma solved a recurrence of order 7 and produced a closed-form evaluation for the example.The result is expressed using multiple harmonic sums and related constants.
  • Initial transformation: TransformToSSums first rewrites the input expression into harmonic sums with rational coefficients and alternating factors.The resulting expression is identified as equation (6.2).
  • Algebraic basis reduction: Equation (6.2) contains 15 different harmonic sums before basis reduction.Algebraic relations from the harmonic-sum algebra are then applied.
  • Algebraic basis reduction: The first reduction yields 11 algebraically independent harmonic sums.The paper states that these representations are algebraically independent.
  • Differential reduction: Differentiation and the operator D = d/dn reduce the representation further to 6 different harmonic sums.The reduction uses relations originating from differentiation.
  • Final reduction: A subsequent half-integer relation leaves 5 multiple harmonic sums together with D and the half-integer function Half.The total number of contributing sums decreases from 15 in (6.2) to 5 in (6.5).
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