What is the i for the S-matrix?

This book provides a modern perspective on the analytic structure of scattering amplitudes in quantum field theory, focusing on the question of whether the S-matrix can be complexified in a way consistent with causality. It demonstrates that causality requirements for individual Feynman diagrams can be mutually incompatible, but that these problems can be addressed by proposing an iε-like prescription for deforming branch cuts without modifying the analytic properties of the physical amplitude. The book is well suited for anyone with knowledge of quantum field theory at a graduate level who wants to become familiar with the complex-analytic structure of Feynman integrals.

Format: Paperback / softback
Length: 165 pages
Publication date: 02 January 2023
Publisher: Springer International Publishing AG

This book offers a contemporary viewpoint on the analytic structure of scattering amplitudes in quantum field theory, with the aim of comprehending and utilizing the implications of unitarity, causality, and locality. It centers on the question: Can the S-matrix be complexified in a manner that is consistent with causality? The affirmative answer has been well understood since the 1960s, in the case of 2→2 scattering of the lightest particle in theories with a mass gap at low momentum transfer, where the S-matrix is analytic everywhere except at normal-threshold branch cuts. We inquire whether an analogous picture extends to realistic theories, such as the Standard Model, that include massless fields, UV/IR divergences, and unstable particles. Especially in the presence of light states running in the loops, the traditional iε prescription for approaching physical regions might break down, because causality requirements for the individual Feynman diagrams can be mutually incompatible. We demonstrate that such analyticity problems are not in contradiction with unitarity. Instead, they should be thought of as finite-width effects that disappear in the idealized 2→2 scattering amplitudes with no unstable particles, but might persist at higher multiplicity. To address these issues, we propose an iε-like prescription for deforming branch cuts in the space of Mandelstam invariants without modifying the analytic properties of the physical amplitude. This procedure results in a complex strip around the real part of the kinematic space, where the S-matrix remains causal. We illustrate all the points on explicit examples, both symbolically and numerically, in addition to providing a pedagogical introduction to the analytic properties of the perturbative S-matrix from a modern point of view. To assist with the i.

Weight: 273g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031182570
Edition number: 1st ed. 2022