Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams

Abstract

By symbolic summation methods based on difference fields we present a general strategy that transforms definite multi-sums, e.g., in terms of hypergeometric terms and harmonic sums, to indefinite nested sums and products. We succeeded in this task with all our concrete calculations of 2--loop and 3--loop massive single scale Feynman diagrams with local operator insertion.

6 pages 2 figures

References in corpus (12)

• Harmonic sums, Mellin transforms and Integrals
• Harmonic Sums and Mellin Transforms up to two-loop Order
• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• Heavy quark coefficient functions at asymptotic values $Q^2\gg m^2$
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5
• A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics
• A Refined Difference Field Theory for Symbolic Summation
• Two-Loop Massive Operator Matrix Elements and Unpolarized Heavy Flavor Production at Asymptotic Values Q^2 >> m^2
• Two--Loop Massive Operator Matrix Elements for Unpolarized Heavy Flavor Production to $O(ε)
• Determining the closed forms of the $O(a_s^3)$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra
• Parameterized Telescoping Proves Algebraic Independence of Sums

Cited by in corpus (54)

• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra
• 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
• The Theory of Deeply Inelastic Scattering
• The O(α_s^3) Massive Operator Matrix Elements of O(n_f) for the Structure Function F_2(x,Q^2) and Transversity
• The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements
• Iterated Binomial Sums and their Associated Iterated Integrals
• The Transition Matrix Element A_{gq}(N) of the Variable Flavor Number Scheme at O(α_s^3)
• 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
• Massive 3-loop Ladder Diagrams for Quarkonic Local Operator Matrix Elements
• 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
• The $O({\alpha }_s^3{n}_f{T}_F^2{C}_{A,F})$} Contributions to the Gluonic Massive Operator Matrix Elements
• A Symbolic Summation Approach to Feynman Integral Calculus
• The Three-Loop Splitting Functions $P_{qg}^{(2)}$ and $P_{gg}^{(2,N_F)}$
• The O(α_s^3 T_F^2) Contributions to the Gluonic Operator Matrix Element
• Creative Telescoping for Holonomic Functions
• The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
• The Heavy Quark Form Factors at Two Loops
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• 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 $gg\to HH$
• 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
• 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
• The two-mass contribution to the three-loop pure singlet operator matrix element
• The Polarized Transition Matrix Element $A_{gq}(N)$ of the Variable Flavor Number Scheme at $O({\alpha }_s^3)$
• 3-Loop Heavy Flavor Corrections to DIS with two Massive Fermion Lines
• The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$
• 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 Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg,Q}$ and $\Delta A_{gg,Q}$
• Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals
• New Heavy Flavor Contributions to the DIS Structure Function $F_2(x,Q^2)$ at $O(α_s^3)
• The Two-mass Contribution to the Three-Loop Polarized Operator Matrix Element $A_{gg,Q}^{(3)}$
• The unpolarized two-loop massive pure singlet Wilson coefficients for deep-inelastic scattering
• Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations
• The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms
• Logarithmic $O({\alpha }_s^3)$ contributions to the DIS Heavy Flavor Wilson Coefficients at $Q^2\gg m^2$
• Massive two-loop four-point Feynman integrals at high energies with AsyInt
• Higher Order Heavy Quark Corrections to Deep-Inelastic Scattering
• Machine Learning Post-Minkowskian Integrals
• General $\epsilon$-representation for scalar one-loop Feynman integrals
• Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d, II: Special kinematics
• Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams
• 3-loop Massive $O(T_F^2)$ Contributions to the DIS Operator Matrix Element $A_{gg}$
• 3-loop heavy flavor Wilson coefficients in deep-inelastic scattering