Mellin-Barnes Integrals

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