Determining the closed forms of the $O(a_s^3)$ anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra

Abstract

Single scale quantities, as anomalous dimensions and hard scattering cross sections, in renormalizable Quantum Field Theories are found to obey difference equations of finite order in Mellin space. It is often easier to calculate fixed moments for these quantities compared to a direct attempt to derive them in terms of harmonic sums and their generalizations involving the Mellin parameter $N$. Starting from a sufficiently large number of given moments, we establish linear recurrence relations of lowest possible order with polynomial coefficients of usually high degree. Then these recurrence equations are solved in terms of d'Alembertian solutions where the involved nested sums are represented in optimal nested depth. Given this representation, it is then an easy task to express the result in terms of harmonic sums. In this process we compactify the result such that no algebraic relations occur among the sums involved. We demonstrate the method for the QCD unpolarized anomalous dimensions and massless Wilson coefficients to 3--loop order treating the contributions for individual color coefficients. For the most complicated subproblem 5114 moments were needed in order to produce a recurrence of order 35 whose coefficients have degrees up to 938. About four months of CPU time were needed to establish and solve the recurrences for the anomalous dimensions and Wilson coefficients on a 2 GHz machine requiring less than 10 GB of memory. No algorithm is known yet to provide such a high number of moments for 3--loop quantities. Yet the method presented shows that it is possible to establish and solve recurrences of rather large order and and degree, occurring in physics problems, uniquely, fast and reliably with computer algebra.

38 pages, 1 style file, 2 FORM files, 2 Mathematica files

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Cited by in corpus (39)

• The Multiple Zeta Value Data Mine
• 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
• A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics
• Large-nf Contributions to the Four-Loop Splitting Functions in QCD
• The 3-Loop Pure Singlet Heavy Flavor Contributions to the Structure Function $F_2(x,Q^2)$ and the Anomalous Dimension
• 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
• Modern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction
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• Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams
• 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 massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_2,F_L,x{F}_3$ and $g_1$
• 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 $O({\alpha }_s^2)$ Heavy Quark Corrections to Charged Current Deep-Inelastic Scattering at large Virtualities
• Three Loop Massive Operator Matrix Elements and Asymptotic Wilson Coefficients with Two Different Masses
• The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
• The Method of Arbitrarily Large Moments to Calculate Single Scale Processes in Quantum Field Theory
• The Heavy Fermion Contributions to the Massive Three Loop Form Factors
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• $O({\alpha }_s)$ Heavy Flavor Corrections to Charged Current Deep-Inelastic Scattering in Mellin Space
• Three loop anomalous dimension of the non-singlet transversity operator in QCD
• The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg,Q}$ and $\Delta A_{gg,Q}$
• The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$
• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals
• The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms
• The inverse Mellin transform via analytic continuation
• The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$
• Higher Order Heavy Quark Corrections to Deep-Inelastic Scattering
• Inverse Mellin Transform of Holonomic Sequences
• Three-Loop Contributions to the Gluonic Massive Operator Matrix Elements at General Values of N
• The QCD Coupling and Parton Distributions at High Precision
• Structure Functions and Low-x -- Working Group Summary