Harmonic Sums and Mellin Transforms up to two-loop Order

1 Introduction

• Mellin transformation maps convolutions into ordinary products, simplifying evolution equations and other hard-scattering calculations.The resulting transforms are genuine quantities in the operator product expansion and can be analytically continued to complex Mellin index N.Source: §1 · p. 2
• The relevant perturbative functions belong to the Nielsen-function class, whose Mellin transforms can be represented as linear combinations of finite harmonic sums.Source: §1 · p. 2
• The paper derives Mellin transforms for all individual functions fi(x) entering massless QED and QCD splitting and coefficient functions through two-loop order.The scope includes time-like processes such as fragmentation and Drell–Yan reactions.Source: §1 · p. 3
• Algebraic relations among harmonic sums reduce the Mellin-transform representations to a compact set of basic functions.The paper contrasts these algebraic representations with linear representations and notes that harmonic-sum structures are easier to classify than Nielsen-function transforms directly.Source: §1 · p. 3
• The paper organizes the analysis around notation, coefficient-function examples, linear harmonic-sum representations, algebraic relations, and explicit appendix transforms.Source: §1 · p. 3

2 Harmonic Sums and Mellin Transforms in Field Theory

• Coefficient and splitting functions are expressed through products of Nielsen functions, whose Mellin transforms generate harmonic sums and higher-transcendentality structures.Source: §2 · pp. 3–4
• Transcendentality assigns degree one to logarithms and degree n to Lin(x), while products and integration measures combine their degrees additively.Source: §2 · pp. 3–4
• Alternating and non-alternating harmonic sums are related to Mellin transforms of Nielsen functions with relatively simple functional or argument structures.Source: §2 · pp. 3–4
• Single harmonic sums are constructed through iterated integrals, providing the basis for more general nested sums.Source: §2 · p. 4
• Up to transcendentality 4, the additional constants are log(2), ζ(2), ζ(3), and Li4(1/2), while higher levels require further constants.Source: §2 · p. 5

3 Coefficient functions at O(α2

• The coefficient functions of FL(x, Q2) are identified as among the simplest two-loop coefficient-function transforms.Source: §3 · p. 5
• All Mellin transforms of the approximately 80 contributing functions fi(x) are tabulated using the linear harmonic-sum representations.Source: §3 · p. 7
• Fourfold sums arising in intermediate steps reduce to more elementary sums, leaving representations involving at most threefold sums.Source: §3 · p. 7
• The resulting expressions assemble splitting functions, coefficient functions, and hard-scattering cross sections for space- and time-like unpolarized and polarized processes through two-loop order.They apply directly for integer Mellin index N; analytic continuations to complex N are given elsewhere.Source: §3 · pp. 7–8
• The paper provides transforms for individual fi(x) rather than only full coefficient functions, extending the results to a wider class of processes.Source: §3 · p. 8
• Agreement was found for the first 20 moments when comparing the c(2)NS coefficient-function representation with Ref..The numbers were also compared with the first 10 moments in Ref. and analytic results in Refs..Source: §3 · p. 8

4 Linear Representations of Harmonic Sums by Mellin Transforms

• Finite harmonic sums through level four are represented as linear combinations of same-level Nielsen-function Mellin transforms and lower-level harmonic sums.The representations arise through consecutive partial integration from integral representations.Source: §4 · p. 8

4.2 Second Order Sums

• Second-order sums include analytic continuations for the simplest cases in terms of ψ^(k)(z) and β^(k)(z).These special-function forms are added alongside the harmonic-sum representations.Source: §4 · p. 8, §4.2 · p. 9

4 Linear Representations of Harmonic Sums by Mellin Transforms

• The section provides explicit linear representations for the harmonic sums and introduces shorthand combinations of Nielsen functions used in those expressions.The notation includes the combination F1(x), which contributes to a series of harmonic sums.Source: §4 · p. 8

4.3 Third Order Sums

• The supplied expressions include log(1 + x) and Li2(x), their product Li2(x) log(1 + x) + 1, and terms combining logarithms with squared log powers.Source: §4.3 · p. 11, §4.4 · pp. 14–15

4.4 Fourth Order Sums

• Comparing linear and algebraic representations may produce convolution relations between Nielsen functions, but the paper does not work these relations out further.This defines the scope boundary of the section’s discussion of such relations.Source: §4.4 · p. 19
• Extensive numerical-precision and computer-algebra checks support the relations, including comparisons of limits with multiple ζ-values when k1 ≠ 1.The corresponding analytic limits are obtained from the integral representation as N approaches infinity.Source: §4.4 · p. 19

5.1 Complete Index Permutation

• Index symmetry produces polynomial combinations of simple harmonic sums whose x-space Mellin transforms reduce to polynomials of convolutions of basic functions.Source: §5.1 · p. 20
• The two-index permutation relation decomposes products of simple sums into ordered sums plus an overlap term determined by the signed indices.Source: §5.1 · p. 20
• Complete index permutations generate 14 relations up to transcendentality four, with additional relations obtained at higher levels.Source: §5.1 · pp. 20–22
• Sums with repeated negative indices admit compact arbitrary-depth forms, while analogous positive-index sums are represented through corresponding recursive structures.Source: §5.1 · pp. 22–23
• Permutation relations provide a first reduction in the number of direct Mellin transforms and relate the transcendental functions appearing in their representations.Source: §5.1 · p. 24

5.2 Partial Permutations and Triple Harmonic Sums

• Partial permutations generalize earlier non-alternating relations to alternating sums and yield relations among triple harmonic sums.Source: §5.2 · p. 24
• Choosing basis elements with simpler Mellin-transform functional structure makes analytic representations for complex N as simple as possible.Source: §5.2 · pp. 24–25
• Algebraic relations reduce threefold linear representations to single harmonic sums and Mellin transforms of a smaller set of functions.Source: §5.2 · pp. 25–26

• All contributing Nielsen-function Mellin transforms for massless QED and QCD splitting functions and Wilson coefficients through two-loop order are calculated and expressed using finite harmonic sums up to level four.Source: §6 · p. 26
• Fourfold level-four sums contribute at two loops only in simplest variants reducible to lower-level sums, but are expected beyond two-loop order.Source: §6 · p. 26
• About 20 basic Mellin transforms, together with known representations expressible through ψ_k(z), suffice to construct the remaining harmonic sums.Source: §6 · pp. 26–27
• Splitting and coefficient functions usually depend on only a subset of the harmonic sums, requiring even fewer basic functions for their Mellin-transform representations.Source: §6 · p. 27

• The appendix includes Mellin-transform terms involving shifted sums such as S_1(N+r), S_2(N+r), and S_3(N+r), including their products.Source: §7 · pp. 28, 32, 36
• The listed formulas also contain mixed products such as log z log^2(1+z) and log^2 z log(1+z).Source: §7 · p. 32
• One representation contains alternating harmonic sums S_1,-2(N−1) and S_2,-1(N−1), alongside products of ordinary sums and log 2.Source: §7 · p. 32
• Several expressions combine z-dependent logarithms, including log z, log(1+z), and their products or powers.Source: §7 · pp. 30, 31, 35, 37