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Computer Algebra Algorithms for Special Functions in Particle Physics

Jakob Ablinger

arXiv:1305.0687v1math-phhep-phhep-th

TL;DR

The paper addresses special nested objects arising in perturbative calculations and develops algorithms for their analysis. It extends properties of harmonic sums and harmonic polylogarithms to generalized objects, including asymptotic analysis and recurrence-based evaluation of integrals.

  • Problem

    Special nested objects arise in perturbative calculations in renormalizable quantum field theories, motivating methods for handling harmonic sums and related quantities.

  • Method

    The paper extends known properties of harmonic sums and harmonic polylogarithms to generalized objects and develops algorithms for relations, series, and asymptotic behavior.

  • Results

    The derived recurrence, together with initial integral values, yields a solution for the evaluated integral.

  • Takeaways & Limitations

    The work provides algorithmic procedures for analyzing generalized nested sums and related integrals.

  • Takeaways & Limitations

    Some integrals could not be processed because of time and space limitations.

Abstract

from arXiv · show

This work deals with special nested objects arising in massive higher order perturbative calculations in renormalizable quantum field theories. On the one hand we work with nested sums such as harmonic sums and their generalizations (S-sums, cyclotomic harmonic sums, cyclotomic S-sums) and on the other hand we treat iterated integrals of the Poincaré and Chen-type, such as harmonic polylogarithms and their generalizations (multiple polylogarithms, cyclotomic harmonic polylogarithms). The iterated integrals are connected to the nested sums via (generalizations of) the Mellin-transformation and we show how this transformation can be computed. We derive algebraic and structural relations between the nested sums as well as relations between the values of the sums at infinity and connected to it the values of the iterated integrals evaluated at special constants. In addition we state algorithms to compute asymptotic expansions of these nested objects and we state an algorithm which rewrites certain types of nested sums into expressions in terms of cyclotomic S-sums. Moreover we summarize the main functionality of the computer algebra package HarmonicSums in which all these algorithms and transformations are implemented. Furthermore, we present application of and enhancements of the multivariate Almkvist-Zeilberger algorithm to certain types of Feynman integrals and the corresponding computer algebra package MultiIntegrate.

DISSERTATION

The dissertation develops computer-algebra methods for nested sums and iterated integrals arising in higher-order renormalizable quantum-field-theory calculations. It derives relations, transformations, asymptotic expansions, and algorithms implemented in HarmonicSums, and applies the multivariate Almkvist–Zeilberger algorithm to selected Feynman integrals through MultiIntegrate.

  • The work studies harmonic sums, S-sums, cyclotomic harmonic sums, cyclotomic S-sums, and related iterated integrals including harmonic, multiple, and cyclotomic harmonic polylogarithms.
  • Generalized Mellin transformations connect the iterated integrals with nested sums, and the dissertation shows how these transformations can be computed.
  • It derives algebraic and structural relations among nested sums and relations between their values at infinity and iterated integrals evaluated at special constants.
  • Algorithms compute asymptotic expansions of the nested objects and rewrite certain nested sums as expressions in cyclotomic S-sums.
  • The HarmonicSums package implements the presented algorithms and transformations.
  • The dissertation presents applications and extensions of the multivariate Almkvist–Zeilberger algorithm to selected Feynman integrals and introduces MultiIntegrate.

5 Cyclotomic S-Sums

This section organizes cyclotomic S-sums through their definitions, algebraic operations, integral and infinity representations, summation procedures, and depth reduction.

  • Cyclotomic S-sums are introduced with their definition and structure.
  • The section treats product and synchronization operations for cyclotomic S-sums.
  • It develops integral representations and values at infinity for cyclotomic S-sums.
  • Summation procedures cover polynomial, rational-function, and two-cyclotomic-S-sum summands.
  • The section also addresses reducing the depth of cyclotomic S-sums.

7 Multi-Variable Almkvist Zeilberger Algorithm and Feynman Integrals 199

This section presents a workflow for applying the multivariate Almkvist–Zeilberger algorithm to Feynman integrals, from recurrence construction through epsilon expansions and package implementation.

  • The algorithm finds a recurrence for the integrand.
  • It then finds a recurrence for the integral.
  • The workflow includes finding epsilon expansions of the integral.
  • The MultiIntegrate package provides the corresponding computer-algebra functionality.

Notations

This notation section defines the principal symbols for harmonic and cyclotomic sums, polylogarithms, Mellin transforms, and associated polynomial spaces and operations.

  • Sa1,a2,...(n) denotes a harmonic sum, while Sa(b; n) denotes an S-sum.
  • Cyclotomic harmonic sums and cyclotomic S-sums are denoted by their indexed S-forms.
  • S(n), C(n), and CS(n) denote polynomial spaces in harmonic sums, cyclotomic harmonic sums, and cyclotomic S-sums, respectively.
  • Hm1,m2,...(x) denotes a harmonic or multiple polylogarithm, and H(m1,n1),(m2,n2),...(x) denotes a cyclotomic harmonic polylogarithm.
  • The notation also defines sign, absolute value, word degree, the Möbius function, and the shuffle product.
  • M(f(x), n) and M+(f(x), n) denote the Mellin and extended Mellin transforms of f(x), respectively.

Introduction

The paper situates harmonic sums and iterated integrals, together with their generalizations, as interconnected objects in perturbative quantum-field-theory calculations. It develops algebraic, structural, transform, and asymptotic tools for these nested objects.

  • Harmonic polylogarithms form a shuffle algebra, while harmonic sums form a quasi-shuffle algebra supporting systematic relations.
  • The paper builds on existing algorithms for Mellin and inverse Mellin transforms, asymptotic expansions, argument transformations, and power-series expansions.
  • Harmonic sums at infinity and harmonic polylogarithms at one are closely related to multiple zeta values and their relations.
  • Generalized harmonic sums and polylogarithms arise in perturbative calculations for massless and massive single-scale problems in quantum field theory.
  • Cyclotomic harmonic sums and S-sums are treated as subsets of the broader class of cyclotomic S-sums.

2 Introduction

The thesis extends algorithms for nested sums and iterated integrals, connecting their transformations, relations, special values, and asymptotics. It implements these methods in HarmonicSums and applies enhanced Almkvist–Zeilberger techniques to Feynman integrals through MultiIntegrate.

  • Nested-sum relations: The thesis derives algebraic and structural relations for finite harmonic sums, S-sums, and cyclotomic harmonic sums, extending earlier harmonic-sum results.The structural relations originate from differentiation and multiplication of the upper summation limit.
  • Iterated integrals: It develops Mellin-type transformations, argument transformations, power-series expansions, and asymptotic algorithms for generalized iterated integrals.
  • Asymptotic analysis: Integral representations yield algorithms for asymptotic expansions of harmonic sums, S-sums, and cyclotomic harmonic sums to arbitrary order.
  • Special values: The work extends relations between values of nested sums at infinity and values of harmonic, multiple, and cyclotomic polylogarithms at special constants.
  • Software: The HarmonicSums Mathematica package implements algorithms for harmonic sums, S-sums, cyclotomic sums, and their associated polylogarithms.
  • Feynman integrals: The thesis extends the multivariate Almkvist–Zeilberger algorithm for certain Feynman integrals and provides MultiIntegrate methods for recurrences, nested-sum representations, and Laurent expansions.The resulting representations use harmonic sums, cyclotomic harmonic sums, and S-sums.

Harmonic Sums

This section develops algebraic structures and transformations for harmonic sums and harmonic polylogarithms, including their Mellin connection, values at special points, and asymptotic expansions. It also provides basis relations and algorithms covering general harmonic sums, including negative indices.

  • Asymptotic expansions: A new approach computes asymptotic expansions of harmonic sums up to arbitrary order, including sums with negative indices.The method uses integral representations, harmonic-sum properties, and extended harmonic polylogarithms closed under the required transformations.
  • Mellin transformations: The chapter presents a new algorithm for computing Mellin transformations of harmonic polylogarithms and relates their values at one to harmonic sums at infinity.The approach differs from an earlier method and supports subsequent algebraic and asymptotic computations.
  • Asymptotic expansions: The asymptotic algorithm extends harmonic polylogarithms by adding a letter so they remain closed under the transformations it requires.The resulting framework supports analytic continuation properties used in the expansion procedure.
  • Algebraic structure: Harmonic-sum products can be rewritten as linear combinations of harmonic sums, forming a quasi-shuffle algebra.The rewriting follows by iterative application of the multiplication identity for sums with the same upper limit.
  • Algebraic structure: Harmonic sums can be decomposed into polynomials in S1(n), with coefficients formed by sums without leading or trailing ones.The extraction strategy repeatedly isolates powers of S1(n) from sums containing leading ones.
  • Values at infinity: The chapter derives relations that reduce basis sums and express harmonic sums at infinity up to weight 8 using 19 constants.The basis relations include algebraic independence results and concrete values summarized through weight 8.

S-Sums

The paper generalizes harmonic sums to S-sums and develops corresponding algebraic, structural, analytic, and asymptotic tools. It connects S-sums with multiple polylogarithms through integral representations and Mellin transforms, while deriving basis reductions, convergence conditions, and algorithms for transformations and expansions.

  • Algebraic structure: S-sums generalize harmonic sums and form a quasi-shuffle algebra, enabling algebraic relations among products of sums.Products with a common upper summation limit can be recursively rewritten as linear combinations of single S-sums.
  • Analytic representation: Integral representations connect S-sums to generalized multiple polylogarithms, extending harmonic polylogarithms beyond the alphabet {−1, 0, 1}.The extension supports structural relations, asymptotic expansions, and identities between polylogarithms at related arguments.
  • Special values: Finite multiple polylogarithms at suitable constants can be rewritten in terms of S-sums at infinity, and this process can be reversed.The paper uses these relations to connect special values of iterated integrals with nested sums.
  • Basis reduction: The paper derives basis reductions for S-sums, including 6 basis sums for 18 sums and 9 basis sums for 19 sums in representative examples.The reductions may introduce additional lower-depth S-sums subject to quasi-shuffle relations.
  • Convergence: Absolute convergence of S-sums at infinity is characterized by three conditions involving the first index and successive products of arguments.The listed cases include |x1| < 1 with bounded successive products, a boundary case with a1 > 1, and a condition involving a1 = 1 and x1 = −1.
  • Algorithms: The paper extends algorithms for harmonic sums to S-sums, including algorithms for asymptotic expansions and transformations involving ¯S-sums.It also generalizes relations for dependent sums and analyzes new basis elements associated with roots of unity.

Cyclotomic Harmonic Sums

The chapter develops the algebraic structure, relations, asymptotics, and Mellin transformations of cyclotomic harmonic sums and related polylogarithms. It also derives basis-counting formulas and connects sum values at infinity with polylogarithms evaluated at one.

  • Algebraic structure: Cyclotomic harmonic sums generalize harmonic sums and form a quasi-shuffle algebra under products with a common upper limit.Products can be rewritten as linear combinations of single cyclotomic harmonic sums.
  • Algebraic structure: Leading and trailing ones can be extracted recursively, decomposing sums into polynomials in linear sums and sums without leading ones.The procedure repeatedly reduces the number of leading ones and also supports trailing-one decompositions.
  • Mellin transformations: Mellin transformations of cyclotomic polylogarithms can be computed to arbitrary weight, including cases weighted by cyclotomic polynomials.The resulting transformations provide representations connecting cyclotomic polylogarithms and cyclotomic harmonic sums.
  • Values at infinity: Cyclotomic harmonic sums converge at infinity if and only if their first index satisfies c1 ≠ 1.This condition determines which sums admit finite values at the infinite upper limit.
  • Values at infinity: Algebraic, duplication, multiple-argument, and generalized harmonic-sum relations remain available when cyclotomic sums are evaluated at infinity.These relations support basis representations of infinite cyclotomic harmonic sums.
  • Asymptotic expansions: The work gives algorithms for asymptotic expansions of cyclotomic sums and develops shifted cyclotomic harmonic polylogarithms as an extension of earlier methods.Specific expansions are combined to obtain results for selected cyclotomic sums, while general linear sums are reduced to S1,...,1(n).

Cyclotomic S-Sums

Cyclotomic S-sums unify S-sums and cyclotomic harmonic sums while supporting quasi-shuffle, synchronization, convergence, and depth-reduction results. The chapter also gives transformations of nested sums into cyclotomic S-sums and reductions through cyclotomic Euler-sums.

  • Cyclotomic S-sums contain S-sums and cyclotomic harmonic sums as subsets.
  • Cyclotomic S-sums form a quasi-shuffle algebra, yielding algebraic product relations.
  • Synchronizing upper summation limits produces multiple-argument and duplication relations, alongside further algebraic relations.
  • Absolute convergence at infinity holds when |x1| < 1 with later cumulative products bounded by 1, or when c1 > 1 and |x1| = 1 with subsequent products bounded by 1.

The Package HarmonicSums

HarmonicSums implements the thesis algorithms for nested sums and iterated integrals, including transformations, expansions, asymptotics, Mellin operations, and differentiation. Its commands demonstrate algebraic expansion and conversions between finite or infinite sums and polylogarithmic representations.

  • HarmonicSums implements algorithms for harmonic sums, S-sums, cyclotomic sums, and several classes of polylogarithms.
  • TransformToSSums rewrites nested sum expressions in terms of harmonic sums, S-sums, cyclotomic harmonic sums, and cyclotomic S-sums.
  • AutoSync[True] synchronizes sums evaluated at a · n + b to argument n.
  • LinearHExpand and LinearExpand expand products of iterated integrals and nested sums into sums of reordered products.
  • HToS and HInfSeries compute series or asymptotic behavior, while HToSinf and SinfToH transform between polylogarithms and sums at infinity.
  • Mellin and InvMellin compute forward and inverse Mellin transformations for supported polylogarithmic and sum expressions.

Multi-Variable Almkvist Zeilberger Algorithm and Feynman Integrals

The chapter applies and enhances the multivariate Almkvist-Zeilberger method to hyperexponential integrands from Feynman-integral problems. MultiIntegrate constructs recurrences through coefficient systems and homomorphic-image speedups, with correctness certificates and example recurrences.

  • The considered Feynman integrals can be mapped to unit-cube integrals whose integrands fit the multivariate Almkvist-Zeilberger framework.
  • MultiIntegrate is an enhanced implementation that computes recurrences for hyperexponential integrands and their integrals.
  • The algorithm represents the integrand as a polynomial times a hyperexponential term and solves coefficient equations for recurrence certificates.
  • The search increases the recurrence order L until a nontrivial solution exists, which is guaranteed for sufficiently large L.
  • Solving the coefficient system is usually the bottleneck for complicated examples, especially with several symbolic parameters.
  • Homomorphic images speed up solving the coefficient system by modular substitutions, null-space computations, and system reduction.
  • The method produces certified homogeneous linear recurrences with polynomial coefficients for example integrals.

2. DIVIDE: As worked out in above, compute a recurrence relation

The method computes recurrences for hyperexponential multi-integrals, recursively simplifies their inhomogeneous parts, and solves the resulting recurrences to express expansion coefficients as indefinite nested product-sums when possible.

  • FLSR: Algorithm FLSR determines whether formal Laurent-series coefficients of the recurrence solution are expressible in terms of indefinite nested products and sums.It returns the maximal computable coefficient range together with expressions valid for all sufficiently large indices.
  • FLSR: The theorem guarantees expressions for the maximal coefficient range only when the required initial values can be computed.The resulting expressions agree with the original coefficients beyond a threshold λ.
  • DIVIDE: A recurrence with polynomial coefficients is derived for the integral, with an inhomogeneous part containing simpler hyperexponential multi-integrals.The simpler integrals have fewer integration quantifiers than the original problem.
  • CONQUER: The CONQUER step applies the strategy recursively to simpler integrals and expands the resulting inhomogeneous term in ε.This produces coefficients h_t(N), …, h_u(N) through order ε^u.
  • Limitations: The approach can fail when nested product-sum expressions cannot be found, and some integrals remain unprocessed because of time and space limits.Computing initial values for all arising integrals is another identified bottleneck.
  • Implementation: The methods are implemented in MultiIntegrate, including direct integration, recurrence-based integration, and ε-expanded integration functions.The package uses Sigma, EvaluateMultiSum, and HarmonicSums as supporting packages.

Curriculum Vitae

The curriculum vitae records Austrian academic training, research appointments, and related study and service experience through 2012.

  • Affiliation: The listed institutional affiliation is the Research Institute for Symbolic Computation at Johannes Kepler University Linz, Austria.
  • Education: The author completed technical mathematics studies at Johannes Kepler University Linz and earned a Diplom Ingenieur with distinction in 2009.The diploma thesis concerned a computer algebra toolbox for harmonic sums related to particle physics.
  • Service: The author served at a Croatian centre for peace, non-violence, and human rights from August 2003 to September 2004.
  • Research experience: The author undertook an exchange at Charles University in Prague and a research stay at DESY in Zeuthen.The exchange occurred in 2007, and the DESY stay ran from September 2011 to April 2012.
  • Education: The author completed doctoral studies at the Research Institute for Symbolic Computation from 2009 to 2012.
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