Computer Algebra Algorithms for Special Functions in Particle Physics

Abstract

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.

233 pages, PhD thesis (2012)

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Cited by in corpus (65)

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• The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
• Numerical Implementation of Harmonic Polylogarithms to Weight w = 8
• The Heavy Fermion Contributions to the Massive Three Loop Form Factors
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
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• The 3-Loop Non-Singlet Heavy Flavor Contributions to the Structure Function g_1(x,Q^2) at Large Momentum Transfer
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• The Polarized Transition Matrix Element $A_{gq}(N)$ of the Variable Flavor Number Scheme at $O({\alpha }_s^3)$
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• 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 $O({\alpha }^2)$ Initial State QED Corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z_0^{\ast }$
• 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
• Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals
• Iterated integrals over letters induced by quadratic forms
• Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations
• Exact N${}^3$LO results for $q{q}^{\prime }\to H+X$
• The real-virtual antenna functions for $S\to Q\bar{Q}X$ at NNLO QCD
• The QED Initial State Corrections to the Forward-Backward Asymmetry of $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z^{0\ast }$ to Higher Orders
• A toolbox to solve coupled systems of differential and difference equations
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• New Results on the 3-Loop Heavy Flavor Corrections in 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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• Inverse Mellin Transform of Holonomic Sequences
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• General $\epsilon$-representation for scalar one-loop Feynman integrals
• Generalizations of polylogarithms for Feynman integrals
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