Mellin-Barnes Integrals. A Primer on Particle Physics Applications

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Preface Acknowledgments Contents Acronyms 1 Precision in Perturbative Particle Physics 1.1 Loops and Real Quantum Corrections in Precision Physics 1.2 Singularities of Amplitudes in QFT 1.3 Dimensional Regularization and Evaluation of Feynman Integrals 1.4 Basic Idea of Mellin-Barnes Representations 1.5 Analytical and Numerical Approaches to Mellin-Barnes Integrals 1.6 Mellin and Barnes Meet Euclid and Minkowski 1.7 A Simple Example as an Invitation to the MB Topic Problems References 2 Complex Analysis 2.1 Complex Numbers and Complex Functions 2.2 The Complex Logarithm 2.3 Generalizations of the Logarithm Function: Classical Polylogarithms 2.3.1 The Arcus Tangent and the Logarithm 2.3.2 The General Dilogarithmic Integral 2.3.3 Classical Polylogarithms 2.4 Multiple Polylogarithms and Beyond 2.5 Gamma Function 2.6 Residues and Cauchy's Residue Theorem 2.7 Gamma and Hypergeometric Functions, Hypergeometric Integrals Problems References 3 Mellin-Barnes Representations for Feynman Integrals 3.1 Representations of Feynman Integrals 3.2 Topological Structure of Feynman Diagrams, Graph Polynomials 3.3 Master Mellin-Barnes Formula: Prescription for the Contour, Proof 3.4 Construction of MB Representations for Feynman Integrals: An Example with Basic Steps 3.5 Simplifying MB Representations 3.6 Using Barnes Lemmas Efficiently 3.7 Loop-by-Loop (LA) Approach 3.7.1 LA Approach, Planar Example 3.7.2 LA Approach, Non-planar Example 3.8 Cheng-Wu Theorem 3.9 Global (GA) Approach 3.9.1 GA Approach and CW Theorem, the Non-planar Double Box 3.9.2 General Two-Loop Skeleton Diagrams 3.9.3 Generalization to Three-Loop Integrals 3.10 Hybrid (HA) Approach 3.11 Minimal Dimensions of Symanzik Polynomials: Summary 3.12 Beyond Three Loops 3.13 MB and the Method of Brackets 3.14 Phase Space MB Integrals Problems References 4 Resolution of Singularities 4.1 Where Do the Poles Come From? 4.2 Resolving Poles: Straight Line and Deformed Contours 4.2.1 Bromwich Contours and Separation of Poles 4.2.2 Analytic Continuation in ε 4.2.3 Analytic Continuation in Auxiliary Parameters Problems References 5 Analytic Solutions 5.1 Residues and Symbolic Summations 5.1.1 Choosing the Contour 5.1.2 From MB Integrals to Sums 5.1.3 Z- and S-Nested Sums 5.1.4 Solving One-Dimensional Mellin-Barnes Integrals with Nested Sums 5.1.5 Application of Sums to Angular Integrations 5.1.6 Expansions of Special Functions 5.1.7 More General Sums and Massive Propagators 5.2 Decoupling Integrals Through a Change of Variables 5.3 Solving via Integration 5.3.1 From Mellin-Barnes to Euler Integrals 5.3.2 Symbolic Integration of Euler Integrals 5.3.3 Merging MB Integrals with Euler Integrals and Symbolic Integration 5.3.4 More General Integrals 5.4 Approximations 5.4.1 Expansions in the Ratios of Kinematic Parameters 5.4.2 Taylor and Region Expansions with MB 5.4.3 MB: Other Directions Problems References 6 MB Numerical Methods 6.1 Introduction 6.2 MB Numerical Evaluation Using Bromwich Contours 6.2.1 Straight-Line Contours and Their Limitations 6.2.2 Transforming Variables to the Finite Integration Range 6.2.3 Shifting and Deforming Contours of Integration 6.2.4 Thresholds and No Need for Contour Deformations 6.3 MB Numerical Evaluation by Steepest Descent 6.3.1 General Idea 6.3.2 Implementation of the Method 6.3.3 Tricks and Pitfalls for ``Vanishing Derivatives'' 6.3.4 Beyond One-Dimensional Cases 6.4 Numerical Evaluation of Phase Space MB Integrals Problems References A Public Software and Codes for MB Studies A.1 Analytic Software A.2 Multiple Sums A.3 Polylogarithms and Generalizations A.4 MB Numerical Software A.5 Other Methods for FI Numerical Integrations B Additional Working Files C Some MB Talks References