[Paper Review] Structural Relations of Harmonic Sums and Mellin Transforms at Weight w=6

J. Blümlein|arXiv (Cornell University)|Jan 7, 2009

Material Science and Thermodynamics22 citations

TL;DR

This paper derives structural relations between harmonic sums and Mellin transforms at weight w=6, focusing on sums without index {-1}, which are relevant for massless QED and QCD calculations. It reduces 486 initial sums to 99, then to 20 basic functions using algebraic and structural relations, enabling analytic continuation to complex N and compact representations of physical quantities like anomalous dimensions and Wilson coefficients.

ABSTRACT

We derive the structural relations between nested harmonic sums and the corresponding Mellin transforms of Nielsen integrals and harmonic polylogarithms at weight {\sf w = 6}. They emerge in the calculations of massless single--scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, to 3-- and 4--loop order. We consider the set of the multiple harmonic sums at weight six without index $\{-1\}$. This restriction is sufficient for all known physical cases. The structural relations supplement the algebraic relations, due to the shuffle product between harmonic sums, studied earlier. The original amount of 486 possible harmonic sums contributing at weight {\sf w = 6} reduces to 99 sums with no index $\{-1\}$. Algebraic and structural relations lead to a further reduction to 20 basic functions. These functions supplement the set of 15 basic functions up to weight {\sf w = 5} derived formerly. We line out an algorithm to obtain the analytic representation of the basic sums in the complex plane.

Motivation & Objective

• To derive structural relations between harmonic sums and Mellin transforms at weight w=6, extending prior work up to w=5.
• To identify a minimal set of basic functions that compactly represent physical quantities in massless QED and QCD at 3– and 4–loop order.
• To enable analytic continuation of harmonic sums to complex N, supporting their use in physical applications.
• To reduce the number of independent harmonic sums at weight w=6 by combining algebraic and structural relations.
• To provide a systematic algorithm for obtaining analytic representations of the basic sums in the complex plane.

Proposed method

• Derives structural relations by analyzing harmonic sums at N and integer multiples/fractions of N, extending their domain to N ∈ ℚ.
• Uses Mellin integral representations of harmonic polylogarithms weighted by 1/(1±z) to connect sums to Mellin transforms.
• Applies integration-by-parts and differentiation relations (d^l/dN^l M[f](N) = M[ln^l(z)f(z)](N)) to derive recurrence and structural constraints.
• Applies the shuffle product algebra to reduce the initial set of 486 harmonic sums (without index {-1}) to 99 independent sums.
• Identifies 20 basic functions by combining algebraic and structural relations, forming a minimal basis for w=6.
• Outlines an algorithm to obtain analytic representations of the basic sums in the complex plane using factorial series and recursion.

Experimental results

Research questions

• RQ1 What are the structural relations between harmonic sums and Mellin transforms at weight w=6, beyond algebraic relations?
• RQ2 How can the number of independent harmonic sums at w=6 be reduced using both algebraic and structural constraints?
• RQ3 Which set of 20 basic functions forms a minimal basis for representing physical quantities at w=6 in massless QED and QCD?
• RQ4 How can the analytic continuation of harmonic sums to complex N be systematically derived for physical applications?
• RQ5 What is the analytic structure of the Mellin transforms of the basic functions, and how do they generalize the ψ-function?

Key findings

• The original set of 486 harmonic sums at weight w=6 reduces to 99 independent sums when excluding index {-1}, which is sufficient for all known physical cases.
• Further reduction using algebraic and structural relations leads to a minimal basis of 20 basic functions, supplementing the 15 basic functions known up to w=5.
• The Mellin transforms of the basic functions are factorial series with singularities only at non-positive integers, generalizing the ψ-function and its derivatives.
• The analytic continuation of harmonic sums to complex N is possible via factorial series representations, with known recursion relations and asymptotic expansions.
• The structural relations allow compact analytic representations of physical quantities such as anomalous dimensions and Wilson coefficients in massless QED and QCD at 3– and 4–loop order.
• The method enables exact analytic representations of Mellin transforms, generalizing previous numerical high-precision approximations.