Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms (1302.0378v1)

Abstract: In recent three--loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short $S$-sums) arise. They are characterized by rational (or real) numerator weights also different from $\pm 1$. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincar\'{e} iterated integrals, also called generalized harmonic polylogarithms. In this regard, we obtain explicit analytic continuations by means of asymptotic expansions of the $S$-sums which started to occur frequently in current QCD calculations. In addition, we derive algebraic and structural relations, like differentiation w.r.t. the external summation index and different multi-argument relations, for the compactification of $S$-sum expressions. Finally, we calculate algebraic relations for infinite $S$-sums, or equivalently for generalized harmonic polylogarithms evaluated at special values. The corresponding algorithms and relations are encoded in the computer algebra package {\tt HarmonicSums}.

• The paper introduces key analytic continuations and asymptotic expansions for generalized harmonic sums, enabling precise mapping to polylogarithms.
• The authors develop novel quasi-shuffle algebraic relations and computational algorithms, implemented in the HarmonicSums package.
• The techniques significantly enhance high-loop QCD calculations and combinatorial evaluations by streamlining complex numerical and analytic methods.

Overview of Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms

The paper "Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms" examines the computational and theoretical framework necessary for effectively dealing with generalized harmonic sums, often referred to as $S$-sums, and their associated polylogarithms within the context of modern Quantum Chromodynamics (QCD) as well as combinatorial mathematics. The paper presents a comprehensive treatment of these mathematical structures through algorithmic implementations and analytic continuations.

Key Concepts and Mathematical Structures

Generalized harmonic sums or $S$-sums are an extension of traditional harmonic sums, incorporating additional complexity through rational or real numerator weights that are not restricted to $\pm 1$. These sums take the form:

$$S_{a_1, a_2, \ldots, a_k}(x_1, x_2, \ldots, x_k; n) = \sum_{n \geq i_1 \geq i_2 \geq \cdots \geq i_k \geq 1} \frac{x_1^{i_1}}{i_1^{a_1}} \cdots \frac{x_k^{i_k}}{i_k^{a_k}}$$

These $S$-sums are instrumental in evaluating Feynman diagrams in high-loop calculations and appear in scenarios involving multiple scales in QCD processes.

Theoretical and Algorithmic Advances

The paper establishes vital analytic continuations and relations:

1. Analytic Continuations: The paper facilitates the understanding and handling of $S$-sums in the complex domain, proposing algorithms for their asymptotic expansions. The connection between Mellin and inverse Mellin transforms is leveraged to map generalized harmonic sums to polylogarithms.
2. Algebraic Relations: Novel algebraic structures, such as quasi-shuffle relations, are explored to describe $S$-sums as an algebraic entity. This assists in expressing products and transformations of sums in terms of known quantities within these established algebras.
3. Special Number Classes: Through the paper of $S$-sums' asymptotic behaviors and limits, the research identifies new classes of special numbers, including those connected to multiple zeta values, extending the comprehension of numeric evaluations.
4. Algorithmic Implementations: The paper presents an implementation within the Mathematica software framework, encapsulated in the HarmonicSums package. This package includes functionalities such as transformation between Z-sums and $S$-sums, Mellin and inverse Mellin transformations, and differentiation with respect to summation indices.

Implications and Applications

The results presented in this paper hold significant implications for both theoretical mathematics and practical high-energy physics computations:

• QCD Calculations: In high-loop calculations in QCD, where generalized harmonic sums frequently occur, these techniques provide a structured approach for handling these sums, thus enabling unambiguous, analytical results.
• Algorithmic Efficiency: The methods and algorithms developed offer computational efficiency, reducing the previously intractable problems to calculations manageable within existing symbolic computation systems.
• Mathematical Advance: The identification of special classes of functions, derived from generalized polylogarithms and $S$-sums, enriches the theory of special functions and potential number-theoretic results.

Conclusion

This work represents a significant enhancement in the suite of tools available for tackling intricate problems in particle physics and combinatorial mathematics. The integration of rigorous analytic methods with accessible computational tools positions these findings as a cornerstone for future developments in computational quantum field theory and related mathematical domains. As researchers further explore these extensions and relations, enhanced algorithms and more profound theoretical insights could emerge, leading to advancements in both functional analysis and practical applications in theoretical physics.