Mellin-Barnes Integrals: A Primer on Particle Physics Applications

ISBN-13: 9783031142710 / Angielski / Miękka / 2022 / 283 str.

Ievgen Dubovyk; Janusz Gluza; Gábor Somogyi

Opis

In this book, the authors discuss the Mellin-Barnes representation of complex multidimensional integrals. Experiments frontiered by the High-Luminosity Large Hadron Collider at CERN and future collider projects demand the development of computational methods to achieve the theoretical precision required by experimental setups. In this regard, performing higher-order calculations in perturbative quantum field theory is of paramount importance. The Mellin-Barnes integrals technique has been successfully applied to the analytic and numerical analysis of integrals connected with virtual and real higher-order perturbative corrections to particle scattering. Easy-to-follow examples with the supplemental online material introduce the reader to the construction and the analytic, approximate, and numeric solution of Mellin-Barnes integrals in Euclidean and Minkowskian kinematic regimes. It also includes an overview of the state-of-the-art software packages for manipulating and evaluating Mellin-Barnes integrals. The book is meant for advanced students and young researchers to master the theoretical background needed to perform perturbative quantum field theory calculations.

Spis treści

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

O autorze

Ievgen Dubovyk did his M.Sc. at Kharkiv Karazin National University, Ukraine. He obtained his Ph.D. at the University of Hamburg in 2019 as a member of a theory group at DESY Zeuthen under the support of the DAAD (Deutscher Akademischer Austauschdienst) foundation. In 2020 he got an assistant professor position at the University of Silesia in Katowice. His research interests focus on the phenomenology of the Standard Model of particle physics, particularly on developing new tools and methods for calculating radiative corrections for various scattering processes relevant to present and future accelerator experiments.

 

Professor Janusz Gluza did his Ph.D. at the University of Silesia in Katowice, Poland, his present workplace. He deals with particle physics and their interactions, publishing papers on precise Standard Model calculations, phenomenology at present and future lepton and hadron colliders, and neutrino physics. Scholarship holder of the Foundation for Polish Science and the Alexander von Humboldt Foundation, a three-year postdoc at the DESY-Zeuthen (Germany). He collaborates with scientists from leading research centers in the world (France, Germany, India, Italy, Switzerland, Spain, USA) and Poland (Institute of Nuclear Physics of the Polish Academy of Sciences in Cracow). For several years involved in cooperation with CERN in Geneva and the Future Circular Collider collaboration. An active member of international scientific networks and principal investigator of several grants of the National Science Center in Poland. From 2014 to 2020 chaired the Katowice branch of the Polish Physical Society, and since 2021 he has been a Polish member of the ECFA panel (European Committee for Future Accelerators).

 

Gábor Somogyi obtained his Ph.D. from the University of Debrecen, Hungary, in 2007 and has subsequently worked as a postdoctoral researcher at the University of Zürich, DESY and CERN. From 2014 to 2021, he was a senior scientific associate at the University of Debrecen, obtaining his habilitation in 2019. Since 2021 he has been a staff member at the Institute for Particle and Nuclear Physics at the Wigner Research Centre for Physics in Budapest, Hungary. His research has focused on theoretical particle physics, particularly on computing precise predictions for high-energy particle collision processes relevant for interpreting experimental data gathered at particle colliders. He has lectured on the basics of particle physics and quantum field theory, as well as on advanced topics such as modern mathematical method for computing Feynman integrals.