A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics

Abstract

In this work we present the computer algebra package HarmonicSums and its theoretical background for the manipulation of harmonic sums and some related quantities as for example Euler-Zagier sums and harmonic polylogarithms. Harmonic sums and generalized harmonic sums emerge as special cases of so-called d'Alembertian solutions of recurrence relations. We show that harmonic sums form a quasi-shuffle algebra and describe a method how we can find algebraically independent harmonic sums. In addition, we define a differentiation on harmonic sums via an extended version of the Mellin transform. Along with that, new relations between harmonic sums will arise. Furthermore, we present an algorithm which rewrites certain types of nested sums into expressions in terms of harmonic sums. We illustrate by nontrivial examples how these algorithms in cooperation with the summation package Sigma support the evaluation of Feynman integrals.

References in corpus (10)

• The Three-Loop Splitting Functions in QCD: The Non-Singlet Case
• Harmonic Polylogarithms
• HPL, a Mathematica implementation of the harmonic polylogarithms
• The third-order QCD corrections to deep-inelastic scattering by photon exchange
• Harmonic Sums and Mellin Transforms up to two-loop Order
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5
• Analytic Continuation of Mellin Transforms up to two-loop Order
• Analytic Continuation of the Harmonic Sums for the 3--Loop Anomalous Dimensions
• Determining the closed forms of the $O(a_s^3)$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra

Cited by in corpus (75)

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• The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity
• The 3-Loop Pure Singlet Heavy Flavor Contributions to the Structure Function $F_2(x,Q^2)$ and the Anomalous Dimension
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• Computer Algebra Algorithms for Special Functions in Particle Physics
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• Iterated Binomial Sums and their Associated Iterated Integrals
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• The Transition Matrix Element A_{gq}(N) of the Variable Flavor Number Scheme at O(α_s^3)
• Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable
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• Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams
• Determining the closed forms of the $O(a_s^3)$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra
• The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements
• Quantum Spectral Curve for a Cusped Wilson Line in N=4 SYM
• The Complete $O({\alpha }_s^2)$ Non-Singlet Heavy Flavor Corrections to the Structure Functions $g_{1,2}^{ep}(x,Q^2)$, $F_{1,2,L}^{ep}(x,Q^2)$, $F_{1,2,3}^{\nu (\bar{\nu })}(x,Q^2)$ and the Associated Sum Rules
• The Three-Loop Splitting Functions $P_{qg}^{(2)}$ and $P_{gg}^{(2,N_F)}$
• A Symbolic Summation Approach to Feynman Integral Calculus
• The $O({\alpha }_s^2)$ Heavy Quark Corrections to Charged Current Deep-Inelastic Scattering at large Virtualities
• The O(α_s^3 T_F^2) Contributions to the Gluonic Operator Matrix Element
• Three Loop Massive Operator Matrix Elements and Asymptotic Wilson Coefficients with Two Different Masses
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
• The Heavy Quark Form Factors at Two Loops
• Numerical Implementation of Harmonic Polylogarithms to Weight w = 8
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• The Heavy Fermion Contributions to the Massive Three Loop Form Factors
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• Heavy Quark Form Factors at Three Loops in the Planar Limit
• The 3-Loop Non-Singlet Heavy Flavor Contributions to the Structure Function g_1(x,Q^2) at Large Momentum Transfer
• The O(α_s^3) Heavy Flavor Contributions to the Charged Current Structure Function xF_3(x,Q^2) at Large Momentum Transfer
• On the relation between the $\overline{\mathrm{MS}}$ and the kinetic mass of heavy quarks
• The three-loop single mass polarized pure singlet operator matrix element
• MPL - a program for computations with iterated integrals on moduli spaces of curves of genus zero
• Massive planar and non-planar double box integrals for light Nf contributions to gg->tt
• The two-mass contribution to the three-loop pure singlet operator matrix element
• P-wave contribution to third-order top-quark pair production near threshold
• The Polarized Transition Matrix Element $A_{gq}(N)$ of the Variable Flavor Number Scheme at $O({\alpha }_s^3)$
• The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$
• The $O({\alpha }^2)$ Initial State QED Corrections to $e^{+}{e}^{-}$ Annihilation to a Neutral Vector Boson Revisited
• Subleading Logarithmic QED Initial State Corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }/{Z^0}^{\ast }$ to $O({\alpha }^6{L}^5)$
• The Polarized Two-Loop Massive Pure Singlet Wilson Coefficient for Deep-Inelastic Scattering
• The three-loop polarized pure singlet operator matrix element with two different masses
• The $O({\alpha }^2)$ Initial State QED Corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z_0^{\ast }$
• The $N_f{C}_F^3$ contribution to the non-singlet splitting function at four-loop order
• The Two-mass Contribution to the Three-Loop Polarized Operator Matrix Element $A_{gg,Q}^{(3)}$
• The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$
• Iterated integrals over letters induced by quadratic forms
• Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums
• The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms
• Exact N${}^3$LO results for $q{q}^{\prime }\to H+X$
• The QED Initial State Corrections to the Forward-Backward Asymmetry of $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z^{0\ast }$ to Higher Orders
• The real-virtual antenna functions for $S\to Q\bar{Q}X$ at NNLO QCD
• A toolbox to solve coupled systems of differential and difference equations
• High Precision QED Initial State Corrections for $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z^{\ast }$ Annihilation
• New Results on the 3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering
• Inverse Mellin Transform of Holonomic Sequences
• Higher Order Heavy Quark Corrections to Deep-Inelastic Scattering
• Charm-quark mass effects in NRQCD matching coefficients and the leptonic decay of the ${\rm Y}(1S)$ meson
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• Three loop QCD corrections to heavy quark form factors
• NNLO mixed EW-QCD corrections to single vector boson production
• 3-loop heavy flavor Wilson coefficients in deep-inelastic scattering
• Recent progress on the calculation of three-loop heavy flavor Wilson coefficients in deep-inelastic scattering