Johannes Blümlein - Anti-Differentiation and the Calculation of Feynman Amplitudes

Mathematics is a key technology in modern society. Symbolic Computation is on its way to become a key technology in mathematics. Texts and Monographs in Symbolic Computation provides a platform devoted to reflect this evolution. In addition to reporting on developments in the field, the focus of the series also includes applications of computer algebra and symbolic methods in other subfields of mathematics and computer science, and, in particular, in the natural sciences. To provide a flexible frame, the series is open to texts of various kind, ranging from research compendia to textbooks for courses.

Indexed by zbMATH.

More information about this subseries at http://www.springer.com/series/3073

This Springer imprint is published by the registered company Springer Nature Switzerland AG

The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

The analytic calculation of Feynman amplitudes, and in particular the evaluation of Feynman integrals, is a challenging task within elementary particle physics. In particular, the simplification of such often large-scale expressions in terms of special functions and constants plays a decisive role to improve the prediction and to obtain a deeper understanding of these predictions of renormalizable quantum field theories of elementary particle physics for experimental precision measurements. The main focus of the book Anti-Differentiation and the Calculation of Feynman Amplitudes is put on sophisticated and efficient symbolic methods and related tools from mathematics and theoretical physics to support this task. Special focus is put on the manipulation and simplification of non-trivial Feynman integrals, and more generally of rather general classes of multi-integrals and connected multi-sums that are relevant in many other research areas of natural and technical science.

This book brings together sophisticated tools from pure mathematics, computer algebra, and the theoretical particle physics community that are combined non-trivially or that have strong potentials for future interaction. In particular, we expect that the following tools presented in this book will be crucial for challenging calculations that will be fundamental for ongoing and future experiments at the Large Hardron Collider (LHC) and its planned successor at CERN, the FCC.

In Analytic Integration Methods in QFT (Johannes Blmlein), a general overview of the mathematical and symbolic computation methods for the treatment of Feynman integrals is elaborated. In particular, the chapters of this book are introduced accordingly in this general presentation.

In Extensions of the AZ-algorithm and the Package MultiIntegrate (Jakob Ablinger) an extension of the (continuous) multivariate Almkvist-Zeilberger algorithm is introduced that can compute linear difference and differential equations of Feynman integrals. Special focus is also put on solving the found equations in terms of nested sums and integrals.

In Empirical Determinations of Feynman Integrals Using Integer Relation Algorithms (Kevin Acres and David Broadhurst) efficient algorithms (in particular, PSLQ and LLL) are presented and applied to evaluate and explore sophisticated Feynman integrals in terms of special constants.

In N=4 SYM Gauge Theories: The 26 Amplitude in the Regge Limit (Jochen Bartels) the Regge limit of scattering amplitudes is utilized, which is based upon unitarity, energy discontinuity, and the analytic structure of the arising Feynman amplitudes.

In Direct Integration for Multi-Leg Amplitudes: Tips, Tricks, and When They Fail (Jacob L. Bourjaily, Yang-Hui He, Andrew J. McLeod, Marcus Spradlin, Cristian Vergu, Matthias Volk, Matt von Hippel, and Matthias Wilhelm) it is demonstrated how the hyperlogarithmic integration method in interaction with new techniques can be used to transform certain Feynman integrals to hyperlogarithms.