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Algebraic Relations Between Harmonic Sums and Associated Quantities

J. Blümlein

arXiv:hep-ph/0311046v1hep-phhep-lathep-thmath.COmath.QAmath.RA

TL;DR

Finite harmonic sums are central to Mellin-space expressions in higher-order QED and QCD, but their rapidly growing number makes analytic representations complex. The paper derives algebraic relations through depth 6, shows that independent sums are counted by Lyndon words, and finds that their fraction is at most 1/d through depth 10, while noting that complete expressions become unwieldy for some fivefold sums.

  • Problem

    The growing number of finite harmonic sums complicates analytic continuation and the representation of higher-order Mellin-space results.

    Source: §1 · p. 2

  • Method

    The paper derives product-based algebraic relations through depth 6 and counts independent sums using Lyndon words associated with each index set.

    Source: §1 · p. 3, §8 · p. 28

  • Results

    The fraction of basic sums is always ≤1/d compared with all sums for the tabulated cases through depth d = 10.

    Source: §8 · p. 29

  • Takeaways & Limitations

    The relations substantially reduce the functions needed for higher-order massless QED and QCD calculations and extend to other objects sharing the same multiplication relation.

    Source: §8 · p. 29

    For fivefold sums, dependent expressions and coefficient matrices become voluminous, so the paper gives complete results only for simpler cases and representative relations for complicated ones.

    Source: §5 · p. 18

Abstract

from arXiv · show

We derive the algebraic relations of alternating and non-alternating finite harmonic sums up to the sums of depth~6. All relations for the sums up to weight~6 are given in explicit form. These relations depend on the structure of the index sets of the harmonic sums only, but not on their value. They are therefore valid for all other mathematical objects which obey the same multiplication relation or can be obtained as a special case thereof, as the harmonic polylogarithms. We verify that the number of independent elements for a given index set can be determined by counting the Lyndon words which are associated to this set. The algebraic relations between the finite harmonic sums can be used to reduce the high complexity of the expressions for the Mellin moments of the Wilson coefficients and splitting functions significantly for massless field theories as QED and QCD up to three loop and higher orders in the coupling constant and are also of importance for processes depending on more scales. The ratio of the number of independent sums thus obtained to the number of all sums for a given index set is found to be $\leq 1/d$ with $d$ the depth of the sum independently of the weight. The corresponding counting relations are given in analytic form for all classes of harmonic sums to arbitrary depth and are tabulated up to depth $d=10$.

1 Introduction

Finite harmonic sums arise throughout Mellin-space calculations in perturbative QED and QCD, where their algebraic relations can reduce the number of functions needed. This paper derives explicit relations through depth 6 and connects them to broader classes of mathematical objects sharing the same multiplication structure.

Motivation

  • Finite harmonic sums occur in higher-order QCD splitting functions, Wilson coefficients, and amplitudes for several high-energy scattering processes.They also arise in expansions of higher transcendental functions appearing in loop integrals.Source: §1 · p. 2

Mathematical setting

  • Finite harmonic sums connect Mellin transforms with harmonic polylogarithms, Nielsen integrals, Euler–Zagier sums, and nested Z-sums.In the limit N →∞, convergent sums yield multiple zeta values.Source: §1 · p. 2

Motivation

  • Because analytic continuation requires many harmonic sums, algebraic relations can represent all sums of a given weight and depth using an independent basis.The number of sums grows rapidly with depth, motivating systematic reduction before inverse Mellin transformation.Source: §1 · p. 2

Contribution

  • The paper derives general algebraic relations for finite harmonic sums through depth 6 and gives explicit representations through weight 6.The results are intended to cover anomalous dimensions and massless coefficient functions through O(α^3).Source: §1 · p. 3

Scope

  • The relations depend on index-set structure rather than numerical sum values, so they also apply to mathematical objects obeying the same multiplication relation, including harmonic polylogarithms.The paper states that integration-by-parts relations are contained within the product relations when all allowed index permutations are considered.Source: §1 · p. 3

2 General Algebraic Relations

The paper derives algebraic relations for finite harmonic sums through depth 6 using product, shuffle, integration-by-parts, and complete-permutation constructions.

2.1 Product–Relations

  • Products of finite harmonic sums reduce sums of a given depth to linear combinations of sums at the same or lower depth.The reduction follows from recursive product relations and shuffle products over allowed index permutations.Source: §2.1 · p. 4
  • Binary shuffle products suffice for the genuine harmonic-sum relations through depth 6; higher multiple shuffles are linearly dependent.These relations are relevant to massless three-loop calculations.Source: §2.1 · pp. 6–7

2.2 Integration-by-Parts Relations

  • Harmonic-polylogarithm relations follow from the harmonic-sum relations by removing terms containing the ∧ contraction symbol.Harmonic polylogarithms obey the shuffle product, while harmonic sums additionally contain contraction terms.Source: §2.2 · p. 8
  • Integration-by-parts relations introduce no new independent relations beyond the product relations for harmonic polylogarithms or finite harmonic sums.The paper establishes this through explicit checks and the Mellin-transform connection between the two classes.Source: §2.2 · pp. 8–9

2.3 Complete Permutations of the Index–Set

  • Complete permutations of a harmonic-sum index set can be represented as polynomials in single harmonic sums, enabling analytic continuation in N.The general construction weights each l1-fold contraction by (l1 −1)! and combines the resulting symmetric-polynomial structures.Source: §2.3 · pp. 9–10
  • The complete-permutation construction extends explicitly to sums of depth 5 and 6 and generates k-fold sums with a single index.The notation distinguishes all permutations from permutations restricting reuse of contracted indices.Source: §2.3 · pp. 9–10

3 The threefold sums revisited

The paper revisits algebraic relations among threefold finite harmonic sums, showing that index permutations yield an additional relation and reduce the sums to smaller bases.

3 The threefold sums revisited

  • The resulting relations contain the earlier threefold relations for distinct indices, while extending them to alternating finite harmonic sums.Source: §3 · p. 11
  • The explicit relations for threefold sums are algebraic combinations of permuted sums, products, and contracted-index sums.Source: §3 · p. 11
  • The six-equation permutation system has rank 4, producing one more relation than earlier results for threefold sums.Source: §3 · p. 11
  • All threefold sums with three different indices can be expressed using two chosen sums and polynomials of lower-rank harmonic sums.Source: §3 · p. 11
  • The algebraic reduction lowers the required set of basic Mellin-transform functions to at most 23 functions at two-loop order.Source: §3 · p. 12

4 The Fourfold Harmonic Sums

The fourfold analysis classifies harmonic sums by index multiplicity and derives algebraic reductions for each class, including explicit rank-based counts and Mellin-transform representations.

4.1 Harmonic Sums with 4 Different Indices

  • For four distinct indices, 24 sums have a coefficient matrix of rank 18, leaving six basic sums.Source: §4.1 · p. 13
  • Fourfold sums with four distinct indices first occur at weight 6.Source: §4.1 · p. 14

4.2 Harmonic Sums with 3 Different Indices

  • For fourfold sums with three different indices, 12 sums are governed by a rank-9 matrix, so three sums can serve as a basis.Source: §4.2 · p. 14

4.3 Harmonic Sums with 2 Different Indices

  • With two distinct indices, all sums in the class can be expressed by one sum of the same depth and weight.Source: §4.3 · p. 16

4.3.2 Index-Set {a, a, a, b}

  • For the index set {a, a, a, b}, four sums have a rank-3 coefficient matrix, so all are represented in terms of one.Source: §4.3.2 · p. 17
  • The associated weight-4 Mellin transforms can be expressed through polynomials of ψk(N) and β^(k)(N) functions.Source: §4.3.2 · p. 17

5 The Fivefold Harmonic Sums

The fivefold section tabulates index classes and independent-sum counts, while noting that complete dependent relations become lengthy for the more complicated classes.

5 The Fivefold Harmonic Sums

  • The classes Sa,a,b,c,d(N) and Sa,b,c,d,e(N) first contribute at weights 7 and 9, respectively, and only their independent-sum counts are listed.Source: §5 · p. 18

5.1.1 Index-Set {a, a, b, b, c}

  • For the index set {a, a, b, b, c}, 30 sums reduce to six independent sums.Source: §5.1.1 · p. 18

5.1.2 Index-Set {a, a, a, b, c}

  • For {a, a, a, b, c}, 20 sums reduce to four independent sums.Source: §5.1.2 · p. 19

5.2.1 Index-Set {a, a, a, b, b}

  • For {a, a, a, b, b}, ten sums are represented in terms of two because the coefficient matrix has rank 8.Source: §5.2.1 · p. 19

5.2.2 Index-Set {a, a, a, a, b}

  • For {a, a, a, a, b}, five sums have a rank-4 coefficient matrix and are reduced through explicit relations.Source: §5.2.2 · p. 20

6 The Sixfold Harmonic Sums

The sixfold analysis organizes sums into basic index classes and derives reductions whose independent counts become especially simple for classes with one exceptional index.

6 The Sixfold Harmonic Sums

  • The sixfold sums are organized into six basic classes, with up to three cases per class.Source: §6 · p. 21

6.1.1 Index-Set {a, a, a, b, b, b}

  • For {a, a, a, b, b, b}, 20 sums are represented by three basic sums.Source: §6.1.1 · p. 21

6.1.2 Index-Set {a, a, a, a, b, b}

  • For {a, a, a, a, b, b}, 15 sums are represented by two basic sums.Source: §6.1.2 · p. 22

6.1.3 Index-Set {a, a, a, a, a, b}

  • For {a, a, a, a, a, b}, six sums have a rank-5 matrix and can all be represented by one basic sum.Source: §6.1.3 · pp. 22–23
  • For general depth d with index set a, a, ..., a, b, the independent fraction is 1/d because the coefficient matrix has rank d − 1.Source: §6.1.3 · pp. 23–24

7 Number of Independent Harmonic Sums and the Number of Lyndon Words

The paper identifies algebraically independent harmonic sums with Lyndon words associated with each index set and develops counting relations for these bases. Explicit examples show how repeated and distinct letters determine the number of independent sums.

Counting independent sums

  • The number of basic harmonic sums for an index set equals the number of associated Lyndon words, whose letters represent the index multiset.The shuffle algebra is freely generated by Lyndon words, making them the basis elements for the corresponding harmonic-sum class.Source: §7 · pp. 24–25

Lyndon-word construction

  • A word is a Lyndon word when it is lexicographically smaller than every suffix, providing a direct construction rule for candidate basis elements.The alphabet is ordered lexicographically, and the word length equals harmonic-sum depth.Source: §7 · pp. 24–25

Counting ratios

  • The basic-to-total sum ratio is primarily controlled by depth, with common nontrivial divisors producing further reductions; for coprime multiplicities it is 1/n.The fully symmetric single-letter case is treated separately because its sums decompose into polynomials of single sums.Source: §7 · pp. 26, 28

Explicit index patterns

  • For two-letter patterns, the number of Lyndon words varies with multiplicities: depth 4 has one for {a, a, b, b}, depth 5 has two for {a, a, a, b, b}, and depth 6 has three for {a, a, b, b, b}.The listed representatives include aabb, aaabb and aabab, and aaabbb, aababb and aabbab, respectively.Source: §7 · p. 27
  • For index sets containing only different letters, Lyndon words have to begin with a; for words of length n, there are (n − 1)! such words, one nth of all possible combinations.They are formed by placing a to the left of the (n − 1)! permutations of b1, . . . , bn−1.Source: §7 · p. 27

Counting relations

  • The paper compares harmonic-sum counts with basic sums determined by algebraic relations, reporting 2·3^(w−1) sums at weight w and cumulative value 3^(w−1).Basic-sum counts as a function of weight w are given as sums over the second Witt formula.Source: §7 · p. 28

8 Conclusions

The paper derives a Lyndon-word basis for finite harmonic sums and explicit relations through depth 6 and weight 6. These relations reduce the number of functions needed in higher-order calculations, with the basic-sum fraction bounded by 1/d through depth 10.

Contributions

  • The paper expresses all harmonic sums through depth 6 and weight 6 using bases counted by Lyndon words, while giving counting relations through depth 10.The relations depend only on index sets and multiplication rules, so they also apply to objects such as harmonic polylogarithms.Source: §8 · p. 28

Counting result

  • The basic-sum fraction is at most 1/d through depth 10 and is mainly determined by depth rather than weight.Common nontrivial divisors of index multiplicities can make the fraction smaller.Source: §8 · p. 29

Practical consequence

  • Applying these relations reduces the function set needed for higher-order massless QED and QCD calculations before analytic continuation and inverse Mellin transformation.Reducing the N-space basis first limits the number of harmonic sums requiring complex continuation.Source: §8 · p. 29

9 Appendix A: Number of Harmonic Sums and Basic Sums for Individual Index-Pattern of Depth 7 to 10

The appendix lists harmonic-sum and basic-sum counts for individual index patterns from depth 7 through depth 10. These patterns enumerate repeated-letter multiplicities used in the general counting framework.

Appendix scope

  • The cited examples span multiplicity patterns from a fully repeated seven-letter multiset through mixed repetitions such as three a’s and two b’s to all-distinct sets of eight or ten letters.Source: §9 · pp. 30, 32

Counting tables

  • The cited passages list the patterns {a, a, b, b, c, c, d}, {a, a, a, a, a, a, b}, {a, a, a, a, a, b, b}, {a, a, a, a, b, b, b}, {a, a, a, a, a, b, c}, and {a, a, a, a, b, b, c}.Source: §9 · p. 30

10 Appendix B: Overview on all Harmonic Sums up to Depth and Weight 6

The appendix overview lists the depth-5 index patterns {1, 1, 1, 1, 1} and {2, 1, 1, 1, 1}.

Overview

  • The overview includes the depth-5 patterns {1, 1, 1, 1, 1} and {2, 1, 1, 1, 1}.Source: §10 · pp. 33–34
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