Harmonic sums, Mellin transforms and Integrals

Abstract

This paper describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. In addition it treats Mellin transforms and the inverse Mellin transformation for functions that are encountered in Feynman diagram calculations. Together with results for the values of the higher harmonic series at infinity the presented algorithms can be used for the symbolic evaluation of whole classes of integrals that were thus far intractable. Also many of the sums that had to be evaluated seem to involve new results. Most of the algorithms have been programmed in the language of FORM. The resulting set of procedures is called SUMMER.

31 pages LaTeX, for programs, see http://norma.nikhef.nl/~t68/summer

Cited by in corpus (289)

• The Three-Loop Splitting Functions in QCD: The Non-Singlet Case
• Harmonic Polylogarithms
• The Three-Loop Splitting Functions in QCD: The Singlet Case
• Numerical evaluation of multiple polylogarithms
• Hadronic top-quark pair production with NNLL threshold resummation
• Automatized analytic continuation of Mellin-Barnes integrals
• HPL, a Mathematica implementation of the harmonic polylogarithms
• The third-order QCD corrections to deep-inelastic scattering by photon exchange
• The Multiple Zeta Value Data Mine
• Conformal Regge theory
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• PolyLogTools - Polylogs for the masses
• The Multithreaded version of FORM
• QCD Analysis of Polarized Deep Inelastic Data and Parton Distributions
• Three-Loop Results for Quark and Gluon Form Factors
• Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS
• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• The Two-Loop Hexagon Wilson Loop in N = 4 SYM
• Deep Inelastic Structure Functions at two Loops
• The Quark Form Factor at Higher Orders
• Soft triple-real radiation for Higgs production at N3LO
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• Mellin Moments of the {$O({\alpha }_s^3$)} Heavy Flavor Contributions to unpolarized Deep-Inelastic Scattering at $Q^2\gg m^2$ and Anomalous Dimensions
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• The three loop on-shell renormalization of QCD and QED
• Dressing and Wrapping
• QCD Analysis of the Polarized World Data
• QCD Analysis of Polarized Deep Inelastic Scattering Data
• Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond
• Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions
• Higher-Order Corrections in Threshold Resummation
• On quartic colour factors in splitting functions and the gluon cusp anomalous dimension
• Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5
• - XSummer - Transcendental Functions and Symbolic Summation in Form
• Towards a fitting procedure for deeply virtual Compton scattering at next-to-leading order and beyond
• Leading singularities and off-shell conformal integrals
• The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case
• Collider Physics at the Precision Frontier
• Single-valued harmonic polylogarithms and the multi-Regge limit
• A Refined Difference Field Theory for Symbolic Summation
• Expansion around half-integer values, binomial sums and inverse binomial sums
• Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra
• Five-Loop Anomalous Dimension of Twist-Two Operators
• On soft singularities at three loops and beyond
• On the Precise Determination of the Fermi Coupling Constant from the Muon Lifetime
• Large-nf Contributions to the Four-Loop Splitting Functions in QCD
• The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity
• On Third-Order Timelike Splitting Functions and Top-Mediated Higgs Decay into Hadrons
• The 3-Loop Pure Singlet Heavy Flavor Contributions to the Structure Function $F_2(x,Q^2)$ and the Anomalous Dimension
• Third-order QCD corrections to the charged-current structure function F_3
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• Two-Loop Massive Operator Matrix Elements and Unpolarized Heavy Flavor Production at Asymptotic Values Q^2 >> m^2
• Analytic Continuation of Mellin Transforms up to two-loop Order
• Higher-spin correlators
• QCD Corrections to Semi-Inclusive Hadron Production in Electron-Positron Annihilation at Two Loops
• Evaluating single-scale and/or non-planar diagrams by differential equations
• The Theory of Deeply Inelastic Scattering
• The Gluonic Operator Matrix Elements at O(α_s^2) for DIS Heavy Flavor Production
• Single soft gluon emission at two loops
• The O(α_s^3) Massive Operator Matrix Elements of O(n_f) for the Structure Function F_2(x,Q^2) and Transversity
• Quasi-symmetric functions and mod p multiple harmonic sums
• On the Next-to-Next-to-Leading Order Evolution of Flavour-Singlet Fragmentation Functions
• The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements
• Drell-Yan lepton-pair production: $q_T$ resummation at N${}^3$LL accuracy and fiducial cross sections at N${}^3$LO
• On the Generalized Harmonic Polylogarithms of One Complex Variable
• Symbolic Expansion of Transcendental Functions
• Two--Loop Massive Operator Matrix Elements for Unpolarized Heavy Flavor Production to $O(ε)
• Approximate N^3LO Higgs-boson production cross section using physical-kernel constraints
• Soft Expansion of Double-Real-Virtual Corrections to Higgs Production at N${}^3$LO
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
• Iterated Binomial Sums and their Associated Iterated Integrals
• Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral
• Next-to-next-to-leading order prediction for the photon-to-pion transition form factor
• The Transition Matrix Element A_{gq}(N) of the Variable Flavor Number Scheme at O(α_s^3)
• NNLO BFKL Pomeron eigenvalue in N=4 SYM
• The massless two-loop two-point function
• Master integrals for massive two-loop Bhabha scattering in QED
• Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable
• Modern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction
• Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms
• Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams
• On non-singlet physical evolution kernels and large-x coefficient functions in perturbative QCD
• Analytic Continuation of the Harmonic Sums for the 3--Loop Anomalous Dimensions
• Massive 3-loop Ladder Diagrams for Quarkonic Local Operator Matrix Elements
• Determining the closed forms of the $O(a_s^3)$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra
• Mellin Moments of the Next-to-next-to Leading Order Coefficient Functions for the Drell-Yan Process and Hadronic Higgs-Boson Production
• Non-Singlet Structure Functions at Three Loops: Fermionic Contributions
• On the next-to-next-to-leading order QCD corrections to heavy-quark production in deep-inelastic scattering
• $O({\alpha }_s^2)$ Timelike Wilson Coefficients for Parton-Fragmentation Functions in Mellin Space
• Heavy-to-heavy quark decays at NNLO
• Nine-Propagator Master Integrals for Massless Three-Loop Form Factors
• The Logarithmic Contributions to the O(α_s^3) Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering
• The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements
• Two-loop amplitudes with nested sums: Fermionic contributions to e+ e- --> q qbar g
• Quantum Spectral Curve for a Cusped Wilson Line in N=4 SYM
• The massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_2,F_L,x{F}_3$ and $g_1$
• Master Integrals for Fermionic Contributions to Massless Three-Loop Form Factors
• Quantum Spectral Curve at Work: From Small Spin to Strong Coupling in N=4 SYM
• The $O({\alpha }_s^3{n}_f{T}_F^2{C}_{A,F})$} Contributions to the Gluonic Massive Operator Matrix Elements
• Two-Loop Formfactors in Theories with Mass Gap and Z-Boson Production
• 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
• Analytical Result for Dimensionally Regularized Massless On-Shell Planar Triple Box
• All order epsilon-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters
• Three Loop Massive Operator Matrix Elements and Asymptotic Wilson Coefficients with Two Different Masses
• RNS derivation of N-point disk amplitudes from the revisited S-matrix approach
• Hexagon Wilson Loop OPE and Harmonic Polylogarithms
• Higgs boson production at the LHC: NNLO partonic cross sections through order $\epsilon$ and convolutions with splitting functions to N${}^3$LO
• On top-pair hadro-production at next-to-next-to-leading order
• Higher Twist Contributions to the Structure Functions F_2^p(x,Q^2) and F_2^d(x,Q^2) at Large x and Higher Orders
• Gravitational scattering at the seventh order in $G$: nonlocal contribution at the sixth post-Newtonian accuracy
• Analytic integration of real-virtual counterterms in NNLO jet cross sections II
• The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
• The Heavy Quark Form Factors at Two Loops
• The Method of Arbitrarily Large Moments to Calculate Single Scale Processes in Quantum Field Theory
• Generalized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers
• Four-loop splitting functions in QCD -- The quark-quark case
• Two-loop QED Operator Matrix Elements with Massive External Fermion Lines
• Towards a fitting procedure to deeply virtual meson production -- the next-to-leading order case --
• Threshold Resummation of the Structure Function F_L
• Periods and Feynman integrals
• Numerical Implementation of Harmonic Polylogarithms to Weight w = 8
• Analytic three-loop Solutions for N=4 SYM Twist Operators
• The Heavy Fermion Contributions to the Massive Three Loop Form Factors
• Towards the NNLO evolution of polarised parton distributions
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• Soft-gluon resummation for boosted top-quark production at hadron colliders
• Mixed QCD-electroweak corrections to W-boson production in hadron collisions
• The FORM project
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• Heavy Quark Form Factors at Three Loops in the Planar Limit
• Revisiting the S-matrix approach to the open superstring low energy effective lagrangian
• Twist-2 at seven loops in planar N=4 SYM theory: Full result and analytic properties
• Six-Loop Anomalous Dimension of Twist-Three Operators in N=4 SYM
• The 3-Loop Non-Singlet Heavy Flavor Contributions to the Structure Function g_1(x,Q^2) at Large Momentum Transfer
• Higgs boson contribution to the leading two-loop Yukawa corrections to $gg\to HH$
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• $O({\alpha }_s^2)$ and $O({\alpha }_s^3)$ Heavy Flavor Contributions to Transversity at $Q^2\gg m^2$
• Nucleon spin structure functions at NNLO in the presence of target mass corrections and higher twist effects
• $O({\alpha }_s)$ Heavy Flavor Corrections to Charged Current Deep-Inelastic Scattering in Mellin Space
• The Kinetic Heavy Quark Mass to Three Loops
• The O(α_s^3) Heavy Flavor Contributions to the Charged Current Structure Function xF_3(x,Q^2) at Large Momentum Transfer
• Absence of pi^2 terms in physical anomalous dimensions in DIS: verification and resulting predictions
• Four-loop splitting functions in QCD -- The gluon-to-quark case
• LSJK - a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions
• Next-to-next-to-leading order calculations for heavy-to-light decays
• The three-loop single mass polarized pure singlet operator matrix element
• On the relation between the $\overline{\mathrm{MS}}$ and the kinetic mass of heavy quarks
• MT: a Mathematica package to compute convolutions
• The two-mass contribution to the three-loop pure singlet operator matrix element
• Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation
• The two-loop vector form factor in the Sudakov limit
• A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the $\overline{\mathrm{MS}}$-pole quark mass relation
• Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
• P-wave contribution to third-order top-quark pair production near threshold
• The one-loop pentagon to higher orders in epsilon
• Three loop anomalous dimension of the non-singlet transversity operator in QCD
• Anomalous dimensions at twist-3 in the sl(2) sector of N=4 SYM
• Leading transcedental contribution to the four-loop universal anomalous dimension in N=4 SYM
• NumExp: Numerical epsilon expansion of hypergeometric functions
• The N${}^3$LO Scheme-invariant QCD Evolution of the Non-singlet Structure Functions \boldmath $F_2^{\mathrm{NS}}(x,Q^2)$ and $g_1^{\mathrm{NS}}(x,Q^2)$
• The Polarized Transition Matrix Element $A_{gq}(N)$ of the Variable Flavor Number Scheme at $O({\alpha }_s^3)$
• Higgs decay into massive b-quarks at NNLO QCD in the nested soft-collinear subtraction scheme
• Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals
• Tuning FORM with large calculations
• BFKL Spectrum of N=4 SYM: non-Zero Conformal Spin
• The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$
• Two-loop coefficient function for DVCS: Vector contributions
• Exact results for $Z_m^{\mathrm{OS}}$ and $Z_2^{\mathrm{OS}}$ with two mass scales and up to three loops
• The $O({\alpha }^2)$ Initial State QED Corrections to $e^{+}{e}^{-}$ Annihilation to a Neutral Vector Boson Revisited
• Fermionic and Scalar Corrections for the Abelian Form Factor at Two Loops
• Next-to-next-to-leading order QCD corrections to the photon's parton structure
• Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering
• Calculation of Massive 2-Loop Operator Matrix Elements with Outer Gluon Lines
• Subleading Logarithmic QED Initial State Corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }/{Z^0}^{\ast }$ to $O({\alpha }^6{L}^5)$
• The axial-vector contributions in two-photon reactions: pion transition form factor and deeply-virtual Compton scattering at NNLO in QCD
• Resummed small-x and first-moment evolution of fragmentation functions in perturbative QCD
• The three-loop polarized pure singlet operator matrix element with two different masses
• The Polarized Two-Loop Massive Pure Singlet Wilson Coefficient for Deep-Inelastic Scattering
• The $O({\alpha }^2)$ Initial State QED Corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z_0^{\ast }$
• Analytic approximations of $2\to 2$ processes with massive internal particles
• The Leading Power Regge Asymptotic Behaviour of Dimensionally Regularized Massless On-Shell Planar Triple Box
• Universal Higher Order QED Corrections to Polarized Lepton Scattering
• The $N_f{C}_F^3$ contribution to the non-singlet splitting function at four-loop order
• The analytic structure and the transcendental weight of the BFKL ladder at NLL accuracy
• Massive three-loop form factors: anomaly contribution
• On the computation of finite bottom-quark mass effects in Higgs boson production
• The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg,Q}$ and $\Delta A_{gg,Q}$
• Complete $N_f^2$ contributions to four-loop pure-singlet splitting functions
• Production of Stoponium at the LHC
• Calculation of Massless Feynman Integrals using Harmonic Sums
• IBP methods at finite temperature
• The Logarithmic Contributions to the Polarized and Operator Matrix Elements in Deeply Inelastic Scattering
• DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory
• The Two-mass Contribution to the Three-Loop Polarized Operator Matrix Element $A_{gg,Q}^{(3)}$
• Hypergeometric representation of a four-loop vacuum bubble
• A new method for taming tensor sum-integrals
• The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$
• The Asymptotic 3-Loop Heavy Flavor Corrections to the Charged Current Structure Functions $F_L^{W^{+}-W^{-}}(x,Q^2)$ and $F_2^{W^{+}-W^{-}}(x,Q^2)$
• The non-planar contribution to the four-loop anomalous dimension of twist-2 operators: first moments in N=4 SYM and non-singlet QCD
• The unpolarized two-loop massive pure singlet Wilson coefficients for deep-inelastic scattering
• On one master integral for three-loop on-shell HQET propagator diagrams with mass
• A fragmentation-based study of heavy quark production
• Iterated integrals over letters induced by quadratic forms
• Higher loop mixed correlators in N=4 SYM
• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals
• Beam functions for $N$-jettiness at N${}^3$LO in perturbative QCD
• $O({\alpha }_s^2$) Polarized Heavy Flavor Corrections}to Deep-Inelastic Scattering at $Q^2\gg m^2$
• Non-planar universal anomalous dimension of twist-two operators with general Lorentz spin at four loops in N=4 SYM theory
• $\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus
• Soft-gluon and Coulomb corrections to hadronic top-quark pair production beyond NNLO
• Review of AdS/CFT Integrability, Chapter III.4: Twist states and the cusp anomalous dimension
• Resummation effects in the bottom-quark fragmentation function
• Mathematical Structure of Anomalous Dimensions and QCD Wilson Coefficients in Higher Order
• The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms
• Twist-2 at five loops: Wrapping corrections without wrapping computations
• The inverse Mellin transform via analytic continuation
• ABJM quantum spectral curve and Mellin transform
• The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$
• Double-logs, Gribov-Lipatov reciprocity and wrapping
• First results for three-loop deep-inelastic structure functions in QCD
• New type of parametrizations for parton distributions
• Renormalization of twist-two operators in covariant gauge to three loops in QCD
• Mathematics for structure functions
• The QCD Splitting Functions at Three Loops: Methods and Results
• The Virtual Photon Structure to the Next-to-next-to-leading Order in QCD
• Non-planar cusp and transcendental anomalous dimension at four loops in N=4 supersymmetric Yang-Mills theory
• Algorithms to Evaluate Multiple Sums for Loop Computations
• Next-to-leading Log Resummation of Scalar and Pseudoscalar Higgs Boson Differential Cross-Sections at the LHC and Tevatron
• The QED Initial State Corrections to the Forward-Backward Asymmetry of $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z^{0\ast }$ to Higher Orders
• First $O({\alpha }_s^3)$ heavy flavor contributions to deeply inelastic scattering
• The Roman harmonic numbers revisited
• Heavy Flavor DIS Wilson coefficients in the asymptotic regime
• Massive two-loop four-point Feynman integrals at high energies with AsyInt
• Three loop MSbar transversity operator anomalous dimensions for fixed moment n <= 8
• Nonsinglet polarized nucleon structure function in infrared-safe QCD
• Three-loop off-forward evolution kernel for axial-vector operators in Larin's scheme
• Integrability and the AdS/CFT correspondence
• The non-singlet kaon fragmentation function from e^+e^- kaon production
• Renormalization of non-singlet quark operator matrix elements for off-forward hard scattering
• The five-gluon amplitude in the high-energy limit
• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• Reflection identities of harmonic sums of weight four
• High Precision QED Initial State Corrections for $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z^{\ast }$ Annihilation
• Charm-quark mass effects in NRQCD matching coefficients and the leptonic decay of the ${\rm Y}(1S)$ meson
• Four-loop large-n_f contributions to the non-singlet structure functions F_2 and F_L
• NNLO coefficient functions of Higgs and Drell--Yan cross sections in Mellin space
• Dispersion Approach in Two-Loop Calculations
• BFKL Eigenvalue and Maximal Alternation of Harmonic Sums
• Special Values of Generalized Polylogarithms
• A New Method for Calculating Differential Distributions Directly in Mellin Space
• The analytic structure of the BFKL equation and reflection identities of harmonic sums at weight five
• FORM facts
• Hypergeometric functions with rational arguments
• Electroweak double-box integrals for Moller scattering
• Symbolic Summation and Higher Orders in Perturbation Theory
• Exact result in N=4 SYM theory: Generalised double-logarithmic equation
• O(α_s^3) contributions to F_L^{Q\bar{Q}}(x,Q^2) for large virtualities
• Nonplanar Four-Loop Anomalous Dimensions of Twist-Two Operators in N=4 Super Yang-Mills Theory: Higher Moment, General Result, and Cusp Anomalous Dimension
• Analytic continuation of harmonic sums: dispersion representation
• Hermitian Separability of BFKL eigenvalue in Bethe Salpeter approach
• Multiloop calculations in HQET
• Single-valued representation of unpolarized and polarized semi-inclusive deep inelastic scattering at next-to-next-to-leading order
• Structural Relations between Nested Harmonic Sums
• Basis transformation properties of anomalous dimensions for hard exclusive processes
• A Family of Supercongruences Involving Multiple Harmonic Sums
• Sudakov Resummations at Higher Orders
• Unitarity methods for Mellin moments of Drell-Yan cross sections
• Reduction of Multiple Harmonic Sums and Harmonic Polylogarithms
• NNLO splitting and coefficient functions with time-like kinematics
• Anomalous dimensions at small spins
• Analytic continuation of harmonic sums with purely imaginary indices near the integer values
• Multiloop Bubbles for hot QCD
• A Mellin Space Program for W^{+/-} and Z^0 Production at NNLO
• QCD two-loop amplitudes for e+e- --> 3 jets: the fermionic contribution
• Higgs boson production at the LHC: fast and precise predictions in QCD at higher orders
• Singlet and non-singlet three-loop massive form factors
• Scattering amplitudes for e^+e^- --> 3 jets at next-to-next-to-leading order QCD
• The QCD Coupling and Parton Distributions at High Precision
• Three loop QCD corrections to heavy quark form factors
• Analytic auxiliary mass flow to compute master integrals in singular kinematics
• Computing Mellin representations and asymptotics of nested binomial sums in a symbolic way: the RICA package
• NNLO QCD corrections to $\gamma \gamma \to Q\bar{Q}$ from Local Unitarity combined with Coulomb resummation and NLO EW effects
• Universal transcendentality limit of BFKL eigenvalue
• Fermionic contribution to the anomalous dimension of twist-2 operators in N=4 SYM theory, critical indices and integrability
• The Mellin moments of deep inelastic structure functions at two loops