Anti-Differentiation and the Calculation of Feynman Amplitudes

Preface
Contents
Contributors
Analytic Integration Methods in Quantum Field Theory: An Introduction
	1 Introduction
	2 Principle Computation Steps for Feynman Diagrams
	3 Symbolic Integration of Feynman Parameter Integrals
		3.1 PSLQ: Zero-Dimensional Integrals
		3.2 Generalized Hypergeometric Functions and Their Extensions
		3.3 The Analytic Mellin–Barnes Technique
		3.4 Hyperlogarithms
		3.5 The Method of Guessing
		3.6 Difference Equations and Summation Methods
		3.7 Differential Equations
		3.8 Multivalued Almkvist-Zeilberger Algorithm
	4 The Function Spaces
		4.1 Nested Sums
		4.2 Iterated Integrals
		4.3 General Properties of Nested Sums and Iterated Integrals
		4.4 Solutions in the Case of Non First Order Factorizable Recurrences and Differential Operators
		4.5 Spaces of Special Numbers
		4.6 Numerical Representations
	5 Precision Goals in Testing the Standard Model
	6 Conclusions
	References
Extensions of the AZ-Algorithm and the Package MultiIntegrate
	1 Introduction
	2 A Fine-Tuned Multi-Variate Almkvist-Zeilberger Algorithm
		2.1 The General Method
		2.2 Dealing with Non-Standard Boundary Conditions
			2.2.1 Dealing with Inhomogeneous Recurrences
			2.2.2 Adapting the Ansatz to Find Homogeneous Recurrences
		2.3 Computing Series Expansions of the Integrals
	3 A Fine-Tuned Continuous Multi-Variate Almkvist-Zeilberger Algorithm
		3.1 The General Method
		3.2 Dealing with Non-Standard Boundary Conditions
			3.2.1 Dealing with Inhomogeneous Differential Equations
			3.2.2 Adapting the Ansatz to Find Homogeneous Differential Equations
		3.3 Computing Series Expansions of the Integrals
	4 Conclusion
	References
Empirical Determinations of Feynman Integrals Using Integer Relation Algorithms
	1 Introduction
	2 PSLQ and LLL
		2.1 LLL
		2.2 Improvement and Parallelization of PSLQ
		2.3 Examples
		2.4 Relations in the Multiple Zeta Value Data Mine
	3 Counterterms at 7 Loops
	4 Periods and Quasi-Periods in Electrodynamics
		4.1 Bessel Moments and Modular Forms
		4.2 Periods and Quasi-Periods for the Laporta Problem
		4.3 Laporta's Intersection Number
	5 Quadratic Relations
		5.1 Quadratic Relations at Weight 6 and Level 24
		5.2 Quadratic Relations at Levels 14 and 34
		5.3 Level 14, at Space-Like Momentum
		5.4 Level 34, with Mass 17-4
	6 Summary
	References
N = 4 SYM Gauge Theories: The 2 →6 Amplitude in the ReggeLimit
	1 Introduction
	2 General Remarks
	3 A Few Features of the 2→4 and 2→5 Amplitudes
		3.1 The 2 →4 Amplitude
		3.2 The 2→5 Amplitude
	4 The 8 Point Function
	5 Summary and Future Steps
	References
Direct Integration for Multi-Leg Amplitudes: Tips, Tricks, and When They Fail
	1 Introduction
	2 Tips and Tricks for a Rational Result
		2.1 Loop-by-Loop Parametrization
		2.2 Momentum Twistors
		2.3 Splitting the Integration Path
	3 Kinematic Square Roots at Symbol Level
	4 Parametric Square Roots: Elliptic and Beyond
		4.1 Scalar Marginal Integrals
		4.2 More General Examples
	5 Conclusions
	References
A Geometrical Framework for Amplitude Recursions: Bridging Between Trees and Loops
	1 Introduction
	2 Genus Zero
		2.1 Iterated Integrals and Multiple Zeta Values
		2.2 Selberg Integrals and Open-String Configuration-Space Integrals
		2.3 Recursion for Open-String Amplitudes at Genus Zero
	3 Genus One
		3.1 Iterated Integrals at Genus One and Elliptic Multiple Zeta Values
		3.2 Elliptic Multiple Zeta Values
		3.3 Generalized Selberg Integrals at Genus One
		3.4 Selberg Recursion at Genus One
		3.5 Recursive Evaluation of Two-Point Open-String Integrals at Genus One
		3.6 Geometric Interpretation
	4 General Framework and Outlook
		4.1 What Does It Need for a General Recursion?
	References
Differential Galois Theory and Integration
	1 Introduction
	2 Examples
		2.1 A First Toy Example
		2.2 A Second Toy Example
		2.3 Integrals via Reducible Systems
		2.4 Example of Situations Involving Reducible Linear Differential Systems
			2.4.1 Operators from Statistical Physics and Combinatorics
			2.4.2 Variational Equations of Nonlinear Differential Systems
	3 Reduced Forms of Linear Differential Systems
		3.1 Ingredient #1: Differential Galois-Lie Algebra
		3.2 Ingredient #2: Lie Algebra Lie(A) Associated to A
		3.3 Linear Differential Systems in Reduced Form
	4 How to Compute a Reduced Form of a Reducible System
		4.1 Shape of the Gauge Transformation
		4.2 The Adjoint Action of the Diagonal
			4.2.1 Isotypical Decomposition
			4.2.2 Intermezzo: Reduction and Rational Solutions
			4.2.3 Reduction on h5 (8-Dimensional Example)
			4.2.4 Reduction on h10 (8-Dimensional Example)
	References
Top-Down Decomposition: A Cut-Based Approach to Integral Reductions
	1 Introduction
	2 Integrand Decomposition and the OPP Method
	3 Integral Reduction and Intersection Theory
	4 The Top-Down Decomposition Approach
	5 Discussion
	References
Hypergeometric Functions and Feynman Diagrams
	1 Introduction
		1.1 Mellin–Barnes Representation, Asymptotic Expansion, NDIM
		1.2 About GKZ and Feynman Diagrams
		1.3 One-Loop Feynman Diagrams
		1.4 Construction of -Expansion
	2 Horn-Type Hypergeometric Functions
		2.1 Definition and System of Differential Equations
		2.2 Contiguous Relations
	3 Examples
		3.1 Holonomic Rank & Puiseux-Type Solution
			3.1.1 Evaluation of Holonomic Rank: The Hypergeometric Function FN
			3.1.2 Puiseux-Type Solution: Hypergeometric Function FT
		3.2 Construction of the -Expansion via Differential Equations: The Appell Function F3
			3.2.1 Notations
			3.2.2 Onefold Iterated Solution
			3.2.3 Boundary Conditions
			3.2.4 The Rational Parametrization: Towards Multiple Polylogarithms
			3.2.5 The Rational Parametrization: Set 1
			3.2.6 The Rational Parametrization: Set 2
			3.2.7 The Rational Parametrization: Set 3
			3.2.8 The Rational Parametrization: Set 4
			3.2.9 The Rational Parametrization: Set 5
			3.2.10 Explicit Construction of Expansion: Integer Values of Parameters
			3.2.11 Construction of -Expansion via Integral Representation
			3.2.12 Relationship to Feynman Diagrams
	4 Conclusion
	References
Differential Equations and Feynman Integrals
	1 Introduction
	2 History
	3 Calculation of Massive Feynman Integrals
		3.1 Basic Massive Two-Loop Integrals
	4 Evaluation of Series
		4.1 Properties of Series
		4.2 Two-Point Examples
		4.3 Three-Point Examples
		4.4 Properties of Massive Diagrams
	5 Modern Technique of Massive Diagrams
		5.1 Canonical Form of Differential Equations
		5.2 Other Approaches
		5.3 Elliptic Structure
	6 Conclusions
	Appendix: Massive Part of J1(q2,m2) in Eq. (5)
	References
Holonomic Anti-Differentiation and Feynman Amplitudes
	1 Introduction
	2 The Holonomic Systems Approach
	3 Particular Values of Hypergeometric Functions
		3.1 Evaluation of a 2F1
		3.2 Evaluations of 3F2 Hypergeometric Functions
		3.3 Finding More 2F1 Identities
	4 Holonomic Integration
	5 Sunrise in Terms of Bessel-K Functions
	References
Outer Space as a Combinatorial Backbone for Cutkosky Rules and Coactions
	1 Introduction
	2 Incidence Hopf Algebras for (Lower) Triangular Matrices
		2.1 Example: Lower Triangular Matrices
	3 Lower Triangular Matrices from the Cubical Chain Complex
		3.1 The Triangle Graph
		3.2 Summing Up
		3.3 The Dunce's Cap
	4 Conclusions
	Appendix 1: The Cubical Chain Complex
	Appendix 2: The Lower Triangular Matrices M(G,To)
	Appendix 3: Summing Orders and Trees
	References
Integration-by-Parts: A Survey
	1 Introduction
	2 Laporta's Algorithm
	3 Conclusions
	References
Calculating Four-Loop Corrections in QCD
	1 Introduction
	2 Space-Like Kinematics
		2.1 Computational Work-Flow
	3 Time-Like Kinematics
		3.1 Inclusive Cross-Sections
		3.2 Semi-Inclusive Cross-Sections
	4 Conclusions
	References
Contiguous Relations and Creative Telescoping
	1 Preamble
	2 Introduction
	3 The Parameterized Gosper Algorithm
		3.1 The Parameterized Gosper Algorithm
	4 Telescoping Contiguous Relations for z≠1 or p≠q+1
		4.1 Telescoping Contiguous Relations for z≠1 and (p,q)=(1,0)
		4.2 Telescoping Contiguous Relations for z≠1 and (p,q)=(2,1)
	5 Proof of Theorem 1
		5.1 Preparatory Lemmas
		5.2 Proof of Theorem 1
		5.3 Connection to Differential Equations
	6 Applications of Theorem 1
	7 A Package for Computing Telescoping Contiguous Relations
		7.1 Computer Discovery and Proof of (67)
		7.2 Computer Discovery and Proof of (73)
	8 Telescoping Contiguous Relations for z=1: Case A
		8.1 Telescoping Contiguous Relations for z=1 and (p,q)=(1,0)
		8.2 Computer Proof of Gauß' 2 F1 Summation
	9 Telescoping Contiguous Relations for z=1: Case B
		9.1 Telescoping Contiguous Relations for z=1 and (p,q)=(2,1)
	10 Telescoping Contiguous Relations for z=1: Case C
		10.1 Telescoping Contiguous Relations for z=1 and (p,q)=(2,1)
		10.2 Telescoping Contiguous Relations for z=1 and (p,q)=(3,2)
	11 Further Applications
		11.1 Generalizing a Theorem by James A. Wilson
		11.2 Non-minimality of Zeilberger Recurrences
		11.3 Creative Symmetrizing Revisited
	12 Conclusion: q-Case
	References
Nested Integrals and Rationalizing Transformations
	1 Introduction
	2 Obtaining Nested Integrals from Nested Sums
		2.1 Generating Functions
		2.2 Mellin Representations
			2.2.1 Computer algebra approaches to parameter integrals
	3 Rationalizing Transformations
		3.1 General Transformations Mapping 0 to 0
		3.2 Real-Valued Square Roots on the Interval [0,1]
			3.2.1 One Square Root
			3.2.2 Two Square Roots
			3.2.3 Three Square Roots
		3.3 Complex-Valued Square Roots on the Interval [0,1]
			3.3.1 Two Square Roots
			3.3.2 Three Square Roots
	References
Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation
	1 Introduction
	2 The Term Algebra SumProd(G)
	3 The Difference Ring Approach for SumProd(G)
		3.1 The Naive Representation in APS-Extensions
		3.2 The Embedding into the Ring of Sequences and R-Extensions
	4 The Representation Problem
		4.1 Representation of Products in R-Extensions
			4.1.1 Algorithmic Tests
			4.1.2 Algorithmic Representations
		4.2 Representation of Sums
			4.2.1 Algorithmic Tests via (Parameterized) Telescoping
			4.2.2 Basic Representations
			4.2.3 Depth-Optimal Representations
	5 The Summation Paradigms
		5.1 Refined Telescoping
		5.2 Parameterized Telescoping (Including Creative Telescoping)
		5.3 Recurrence Solving
	6 Application: Evaluation of Feynman Integrals
	7 Conclusion
	References
Expansion by Regions: An Overview
	1 Historiographical Notes
	2 Geometrical Formulation
	3 Conclusion
	References
Some Steps Towards Improving IBP Calculations and Related Topics
	1 Introduction
	2 Few Legs and Many Loops
	3 A Little Bit About Rstar
	4 Mathematical Aspects
	5 Computer Algebra
	6 Outlook
	References
Iterated Integrals Related to Feynman Integrals Associated to Elliptic Curves
	1 Introduction
	2 Background from Mathematics
	3 Moduli Spaces
	4 Physics
	5 Conclusions
	References