Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5

Johannes Blümlein

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Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5

Johannes Blümlein

arXiv:0901.3106v2Jan 20, 2009hep-phhep-thmath-phmath.AG

TL;DR

The paper asks how Mellin transforms of weighted Nielsen integrals and harmonic sums can be systematically related in single-scale QED and QCD calculations. It derives structural and algebraic reductions through weight five and analytic representations for the resulting complex functions. The physical subset is reduced from 69 sums to 15 basic meromorphic functions with factorial-series, recursion, and asymptotic representations.

Problem
Single-scale QED and QCD calculations require many nested harmonic sums and their Mellin-transform counterparts, motivating relations that reduce this growing set beyond quasi-shuffle relations.
Method
The paper derives structural relations for harmonic sums and weighted Nielsen-integral Mellin transforms, then represents the basic functions through factorial series and complex analytic continuation.
Results
The 69-sum physical subset is reduced algebraically to 30 sums and structurally to 15 basic functions, which are represented as meromorphic functions with recursion and asymptotic relations.
Takeaways & Limitations
At weight five, 15 basic functions suffice for the stated physical applications, while the basis is reported as universal up to isomorphies across many massless and massive two-loop problems.
Takeaways & Limitations
The framework considers one-parameter functions on (0,1), and the map x → x^2 is not closed within the same function class for Nielsen integrals.

Abstract

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We derive the structural relations between the Mellin transforms of weighted Nielsen integrals emerging in the calculation of massless or massive single--scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, and other hard scattering cross sections depending on a single scale. The set of all multiple harmonic sums up to weight five cover the sums needed in the calculation of the 3--loop anomalous dimensions. The relations extend the set resulting from the quasi-shuffle product between harmonic sums studied earlier. Unlike the shuffle relations, they depend on the value of the quantities considered. Up to weight {\sf w = 5}, 242 nested harmonic sums contribute. In the present physical applications it is sufficient to consider the sub-set of harmonic sums not containing an index $i = -1$, which consists out of 69 sums. The algebraic relations reduce this set to 30 sums. Due to the structural relations a final reduction of the number of harmonic sums to {15} basic functions is obtained. These functions can be represented in terms of factorial series, supplemented by harmonic sums which are algebraically reducible. Complete analytic representations are given for these {15} meromorphic functions in the complex plane deriving their asymptotic- and recursion relations. A general outline is presented on the way nested harmonic sums and multiple zeta values emerge in higher order calculations of zero- and single scale quantities.

1 Introduction

The paper studies harmonic sums and their Mellin-transform counterparts in single-scale quantum-field-theory calculations, extending algebraic relations toward analytic continuation and compact functional bases.

Harmonic sums and associated Nielsen-type Mellin integrals arise in massless and massive single-scale calculations, including anomalous dimensions and Wilson coefficients.
The number of harmonic sums grows exponentially with weight, reaching 3^w−1 for all sums up to weight w.
Shuffle and quasi-shuffle relations reduce harmonic sums to basis elements, while additional relations arise from integral representations, symmetric index sets, and functional values.
The paper extends harmonic sums from positive integer indices N to complex N, deriving new relations involving rational arguments and differentiation.
The resulting harmonic sums are meromorphic, admit factorial-series representations, and have analytic recursion and asymptotic relations.
The paper develops structural Mellin-transform relations through weight five and reorganizes the resulting basis for factorial-series representations.

2 The Emergence of Harmonic Sums

The paper traces nested harmonic sums from Feynman-parameter and Mellin–Barnes representations of single-scale quantum-field-theory calculations, then identifies quantity-independent functional bases.

Single-scale processes generate Mellin moments through light-cone, cut-vertex, or partonic descriptions, with kinematic variables constrained to momentum-fraction ranges.
Higher-order or multi-scale extensions enlarge the integration alphabet and generalize harmonic polylogarithms, multiple zeta values, and nested harmonic sums.
Feynman parameters, loop integrations, and Mellin–Barnes representations produce gamma-function structures whose residues yield multiple infinite sums.
Mellin–Barnes methods can generate more sums than the physical results require, although they expose the structures emerging from the integrals.
Expanding gamma functions in the dimensional regulator generates products of single harmonic sums that combine into nested sums.
Harmonic sums at a fixed weight can be represented using a basis independent of the particular single-scale quantity, identified here through weight five.

3 Basic Definitions

The paper defines Mellin transforms for functions and distributions on the unit interval, connects harmonic-polylogarithm transforms to harmonic sums, and notes limits of argument-mapping relations.

The framework considers smooth real-valued functions on (0,1), selected distributions, and plus-prescriptions relevant to field-theoretic calculations.
The Mellin transforms considered are meromorphic functions of N, and integer branches can be analytically continued to complex N.
Mellin transforms of harmonic polylogarithms weighted by 1/(1±x) yield harmonic sums, while delta-derivative distributions produce additional transforms.
Analytic continuation can use rational arguments such as N/2, with denominator factorization and argument mappings supplying relations.
For Nielsen integrals, the argument map x → x^2 does not remain closed within the same function class, limiting direct use of that relation.

4 Classes of Harmonic Sums

Harmonic sums are organized by alternating and non-alternating index alphabets, with rapidly growing counts and algebraic reductions. In the physical applications considered, sums containing index −1 are absent, substantially narrowing the relevant class.

The number of all possible alternating and non-alternating harmonic sums grows exponentially with weight, following the counting relation for the three-letter alphabet {−1, 0, 1}.
At two- and three-loop order in the stated QCD and QED applications, harmonic sums containing index −1 do not contribute.
The no-−1 subclass is counted by a generating function, with starting values N¬{−1}(1) = 1 and N¬{−1}(2) = 3.
Algebraic relations reduce the sums to a-basic subsets, whose counts are tabulated by weight alongside the full and no-−1 classes.
The counting rule for the no-−1 subclass is related partly to non-rooted-tree counts for weights 2 through 7 but differs otherwise.

5 Sums of weight one

Weight-one harmonic sums admit analytic continuation through the ψ-function and related Mellin-transform identities. Rational-argument and differentiation relations reduce all single harmonic sums to one basic function, S1(N).

Weight-one harmonic sums S±k(N) for complex N can be traced to the ψ-function and its derivatives.
The weight-one functions are connected to the first two cyclotomic polynomials after regularization of the Mellin integrals.
Rational-argument and differential relations express the single harmonic sums through ψ-functions and their derivatives, leaving S1(N) as the only basic single harmonic sum.
Mellin-transform representations yield differential identities whose transforms vanish as N →∞ for continuous integrands containing x^N.

6 Sums of weight two

At weight two, shuffle and integral relations connect harmonic sums and their Mellin transforms, leaving a small set of nontrivial functions beyond S1(N). These functions can be represented using β-related functions and Nielsen’s notation.

Among six weight-two harmonic sums, four decompose into polynomials of single sums, while S−1,1(N) and S1,−1(N) remain.
The Mellin transform F1(N) is the first non-trivial transform beyond the single harmonic sum S1(N), although it contains the index −1 excluded from the final stated physics applications.
The identities involving β and ψ functions provide explicit relations such as β(z)β(1 −z) = η(z) + η(1 −z) and β2(z) = ψ′(z) −2η(z).
Euler’s relation connects S1,−1(N) and S−1,1(N) with S−2(N) and S1(N), enabling further Mellin-transform decompositions.
The connected sums imply relations among their Mellin transforms, and the resulting expressions can use β(N), β′(N), ξ1(N), and ξ2(N).

7 Sums of weight three

At weight three, shuffle relations reduce eighteen harmonic sums to six remaining sums represented by Mellin transforms. A selected basis form is chosen to simplify the complex-asymptotic representation, including a no-−1 subset.

Shuffle relations reduce the eighteen weight-three harmonic sums to six remaining sums represented by Mellin transforms.
Two weight-three functions correspond to S−1,1,−1(N) and S1,1,−1(N), which are not algebraically connected.
Relations among Mellin transforms use differentiation and identities involving transforms of logarithmic and dilogarithmic kernels.
The chosen form for F2(N) yields a particularly simple asymptotic representation for complex N with |N| →∞.
The weight-three basic sums without index −1 are specified as the relevant reduced basis for the physical subclass.

8 Sums of weight four

At weight four, shuffle and structural relations substantially reduce the harmonic-sum basis relevant to physical applications, while some Mellin-transform relations do not yield further reductions.

8 Sums of weight four: After algebraic relations, the weight-four sums without index -1 reduce to seven sums.These have index sets (2,1,1), (-2,1,1), (3,1), (-3,1), (-2,2), (4), and (-4).
8 Sums of weight four: 38 of 54 weight-four harmonic sums can be expressed through 16 sums using shuffle relations.
8 Sums of weight four: Symmetric sums S−2,−2(N) and S2,2(N) relate two of the weight-four Mellin transforms.
8 Sums of weight four: A further relation involving the relevant Mellin transforms does not reduce the basis because both transforms already enter identically.
8 Sums of weight four: The function I1(x)/(1 −x) is excluded because it does not occur in single-scale quantities through weight six.

9 Sums of weight five

Weight-five analysis derives representations for higher-depth harmonic sums and identifies the final basic functions needed at this level, including their algebraic and Mellin-transform structure.

9 Sums of weight five: Weight-five representations derive from relations among twofold, threefold, fourfold, and fivefold sums after algebraic relations are assumed.
9 Sums of weight five: Some Mellin transforms are already known, so new cases are reduced through relations, differentiation, and previously defined basic functions.
9 Sums of weight five: For odd k, selected sums substitute Mellin transforms of Nielsen functions using basic functions and their derivatives.For even k, the analogous decomposition applies to one denominator choice but requires relation (7.4) for the other.
9 Sums of weight five: The final weight-five basis consists of Mellin-transform functions built from Li4, S1,3, S2,2, and related lower-weight structures.The listed weight-five functions are Li4(x)/(x ± 1), S1,3(x)/(x + 1), and S2,2(x)/(x ± 1).
9 Sums of weight five: The five-fold sum S1,1,1,1,1(N) decomposes algebraically into a polynomial of single sums.

10 Conclusions

The paper extends quasi-shuffle relations with value-dependent structural relations, reducing the relevant weight-five harmonic sums to a compact set of analytically continuable basic functions.

10 Conclusions: 69 physical harmonic sums without index -1 reduce to 30 through algebraic relations and to 15 through structural relations.
10 Conclusions: The structural relations comprise fractional-argument, integration-by-parts, and differentiation relations that depend on functional values.
10 Conclusions: The 15 basic sums become meromorphic functions with poles at non-positive integers and admit complex-argument recursion and asymptotic representations.
10 Conclusions: The 15 functions suffice for weight-five physical applications such as three-loop anomalous dimensions, while 37 functions are needed at weight six for three-loop Wilson coefficients.
10 Conclusions: Basic functions are reported to be universal up to isomorphies across many massless and massive two-loop problems.

A Algebraic Relations

The appendix lists algebraic relations that express higher-depth harmonic sums without index -1 in terms of a reduced basis and lower-complexity sums.

A Algebraic Relations: The appendix provides explicit algebraic relations for physical single-scale sums without index -1 through weight five.
A Algebraic Relations: Weight-four and weight-five relations reduce sums such as S1,1,±2 and S1,1,±3 to lower-depth sums, products, and constants.
A Algebraic Relations: Additional identities relate permutations of weight-five indices, including S2,−2,1, S−2,1,2, S1,−2,2, and S2,1,2.
A Algebraic Relations: Higher-depth relations express sums including S1,−2,1,1, S1,1,−2,1, and S1,2,1,1 through products and other harmonic sums.
A Algebraic Relations: The Mellin-transform representations use factorial series plus algebraically decomposable remainder terms, yielding meromorphic functions and lower-complexity recursion relations.

B.1 Asymptotic Representations of Factorial Series

The section develops analytic and asymptotic representations for Mellin-transform-related factorial series and basic functions. It combines algebraic reductions, recursion relations, and asymptotic expansions to represent these functions throughout the complex plane.

Analytic continuation: The factorial-series construction yields meromorphic functions in the complex plane, with poles at non-positive integers and analytic continuation to complex arguments.The associated factorial series Ω(z) provides the continuation framework.
Auxiliary functions: The analytic treatment uses ψ-functions, β-functions, Bernoulli numbers, Stirling numbers, and tangent numbers to construct basic-function representations.These auxiliary sequences and functions supply coefficients, recursions, or asymptotic expansions.
Scope boundary: The Mellin-transform approach does not provide asymptotic representations for every harmonic sum; S_1(N) instead contains a logarithmic contribution through ψ(N + 1).This marks a scope boundary for the outlined asymptotic procedure.
Factorial-series reduction: Remainder terms are reduced to factorial series after separating logarithmic contributions associated with leading strings of index 1.The separated terms produce logarithmic growth proportional to ln^k(z).
Recursion relations: Recursion relations connect points in the analytic region to an asymptotic domain |z| ≥ z_asympt with z_asympt approximately 15–20.The recursions are obtained from integral representations of the Mellin transforms.
Asymptotic expansions: Asymptotic representations retain terms through O(1/z^19), targeting double-precision accuracy while allowing higher-order terms for specialized numerical applications.The expansion covers single harmonic sums and the basic functions F_i(z) and F̂_i(z).