Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable

Abstract

We present an algorithm which allows to solve analytically linear systems of differential equations which factorize to first order. The solution is given in terms of iterated integrals over an alphabet where its structure is implied by the coefficient matrix of the differential equations. These systems appear in a large variety of higher order calculations in perturbative Quantum Field Theories. We apply this method to calculate the master integrals of the three--loop massive form factors for different currents, as an illustration, and present the results for the vector form factors in detail. Here the solution space emerging is given by the cyclotomic harmonic polylogarithms and their associated special constants. No special basis representation of the master integrals is needed. The algorithm can be applied as well to more general cases factorizing at first order, which are based on more general alphabets, iterated integrals and associated constants.

arXiv comment39 pages Latex, 19 figures, 2 style files, and anc-files

References in corpus (67)

• Observation of Top Quark Production in Pbar-P Collisions
• Observation of the Top Quark
• Harmonic Polylogarithms
• The four-loop beta-function in Quantum Chromodynamics
• Multiloop integrals in dimensional regularization made simple
• New features of FORM
• Differential Equations for Feynman Graph Amplitudes
• The three-loop QCD $\beta$-function and anomalous dimensions
• Five-Loop Running of the QCD coupling constant
• The Four-Loop QCD Beta-Function and Anomalous Dimensions
• Numerical evaluation of multiple polylogarithms
• Harmonic sums, Mellin transforms and Integrals
• Reduze - Feynman Integral Reduction in C++
• Group theory factors for Feynman diagrams
• Harmonic Sums and Mellin Transforms up to two-loop Order
• The Multiple Zeta Value Data Mine
• Complete Corrections of O(αα_s) to the Decay of the Z Boson into Bottom Quarks
• Physics with e^+e^- Linear Colliders
• Reduze 2 - Distributed Feynman Integral Reduction
• FIESTA 4: optimized Feynman integral calculations with GPU support
• The elliptic dilogarithm for the sunset graph
• The Multithreaded version of FORM
• Quark mass relations to four-loop order
• Numerical Evaluation of Harmonic Polylogarithms
• Analytic treatment of the two loop equal mass sunrise graph
• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA)
• Infrared singularities of QCD amplitudes with massive partons
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Two-loop QCD Corrections to the Heavy Quark Form Factors: the Vector Contributions
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• The three loop on-shell renormalization of QCD and QED
• Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity
• A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics
• FIESTA 2: parallelizeable multiloop numerical calculations
• The kite integral to all orders in terms of elliptic polylogarithms
• A Refined Difference Field Theory for Symbolic Summation
• Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra
• $\overline{\mathrm{MS}}$-on-shell quark mass relation up to four loops in QCD and a general SU$(N)$ gauge group
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• The three-loop cusp anomalous dimension in QCD
• Computer Algebra Algorithms for Special Functions in Particle Physics
• Two-Loop QCD Corrections to the Heavy Quark Form Factors: Anomaly Contributions
• Iterated Binomial Sums and their Associated Iterated Integrals
• Relation between the pole and the minimally subtracted mass in dimensional regularization and dimensional reduction to three-loop order
• Two-Loop QCD Corrections to the Heavy Quark Form Factors: Axial Vector Contributions
• Three-Loop Chromomagnetic Interaction in HQET
• Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms
• A Symbolic Summation Approach to Feynman Integral Calculus
• Decays of Scalar and Pseudoscalar Higgs Bosons into Fermions: Two-loop QCD Corrections to the Higgs-Quark-Antiquark Amplitude
• The Heavy Quark Form Factors at Two Loops
• Numerical Implementation of Harmonic Polylogarithms to Weight w = 8
• A Symbolic Summation Approach to Find Optimal Nested Sum Representations
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• Heavy Quark Form Factors at Three Loops in the Planar Limit
• Four-loop wave function renormalization in QCD and QED
• Analytic results for planar three-loop integrals for massive form factors
• High energy behaviour of form factors
• The asymptotic behavior of the heavy quark form factors at higher order
• "Sixth root of unity" and Feynman diagrams: hypergeometric function approach point of view
• Three-loop QCD corrections and b-quark decays
• Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations
• Summation Theory II: Characterizations of $\mathbf{R}\mathbf{\Pi }{\mathbf{\Sigma }}^{\ast }$-extensions and algorithmic aspects
• Heavy quark form factors in the large ${\beta }_0$ limit
• From modular forms to differential equations for Feynman integrals
• Three-loop effective potential of general scalar theory via differential equations

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• Five-Loop Static Contribution to the Gravitational Interaction Potential of Two Point Masses
• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• 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 $g_1$
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
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• The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
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• Towards muon-electron scattering at NNLO
• Subleading Logarithmic QED Initial State Corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }/{Z^0}^{\ast }$ to $O({\alpha }^6{L}^5)$
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• Modular transformations of elliptic Feynman integrals
• Massive three-loop form factors: anomaly contribution
• 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
• Identifying regions in wide-angle scattering via graph-theoretical approaches
• The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms
• The inverse Mellin transform via analytic continuation
• Cuts and Isogenies
• The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$
• 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
• Analytical solution to DGLAP integro-differential equation via complex maps in domains of contour integrals
• Sunrise integral with two internal masses and pseudo-threshold kinematics in terms of elliptic polylogarithms
• Three-loop effective potential of general scalar theory via differential equations
• Electroweak double-box integrals for Moller scattering
• Algorithm to find an all-order in the running coupling solution to an equation of the DGLAP type
• NNLO mixed EW-QCD corrections to single vector boson production
• Singlet and non-singlet three-loop massive form factors
• Three loop QCD corrections to heavy quark form factors
• Analytic auxiliary mass flow to compute master integrals in singular kinematics
• A Feynman integral depending on two elliptic curves
• Two-loop mixed QCD-EW corrections to neutral current Drell-Yan