Foreword by Tord Riemann
1. Introduction
- Theory versus experiments: Precision calculations and needs for new methods and tools in perturbative QFT.
- Heart of the problems: singularities of integrals in QFT.
- Dimensional regularization, renormalization, types of instabilities (IR, UV, collinear, thresholds).
- Virtual Feynman integrals, real phase space integrals.
- Basic idea of Mellin-Barnes representations.
- Mellin and Barnes meet Euclid and Minkowski (analytical and numerical solutions of integrals in Euclidean and Minkowskian space).
- Simple worked examples as an "invitation" to the topic.
2. Complex analysis
- Power of complex numbers and complex functions in physics; basic terminology, illustrations.
- Residues and Cauchy's theorem, working examples.
- Complex functions of interest: (Poly)logarithms and Gamma functions. Denitions, properties, analytic structure (poles, behaviour at innity), series expansion. Computing examples.
3. Mellin-Barnes representations for Feynman and related integrals
- Topological structure of Feynman diagrams, loop computations: U, F polynomials. Computing examples.
- Master Mellin-Barnes formula: prescription for the contour, proof.
- Construction of Mellin-Barnes representations for Feynman virtual integrals: loop-by-loop, global and hybrid methods, method of brackets, computing examples.
- Phase space integrals: angular integrals, obtaining MB representations, computing examples.
- Simplifying MB representations: Barnes' lemmas and corollaries, Cheng-Wu theorem, computing examples.
4. Resolution of singularities
- Where do the poles come from?
- Resolving poles: straight line and deformed contours, auxiliary regularization.
- Expanding special functions, analytic continuation.
- Computing examples.
5. Analytic solutions
- Residues and symbolic summations.
- Decoupling integrals through a change of variable.
- Solving via integration: \standard" form, Euler integrals.
- Classes of solved functions: generalized/harmonic polylogarithms, elliptic functions and beyond.
- Tricks and pitfalls, examples.
6. Approximations
- Expansions in the MB variables.
- Expansions in the ratios of kinematic parameters.
- Analytic continuation and summations of the dimensionally reduced MB integrals.
- Tricks and pitfalls, examples.
7. Numerical methods
- Straight line contours and their limitations.
- Transforming variables to the nite integration range, shifting and deforming contours of integration, steepest descent and Lefschetz thimbles, quasi Monte Carlo integrations.
- Modern developments: state-of the-art and possible directions.
- Tricks and pitfalls, examples.
8. Appendix
- Public software and codes.
- More on special functions: 2F1 and generalizations, polylogarithms.
- More on multiple sums, Z- and S-sums, summation algorithms, table of sums.
Glossary
Bibliography