Mellin-Barnes Integrals

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