This book provides a modern perspective on the analytic structure of scattering amplitudes in
quantum field theory with the goal of understanding and exploiting consequences of unitarity
causality and locality. It focuses on the question: Can the S-matrix be complexified in a way
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
ask 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 i 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 fix these issues we propose an i -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 giving a pedagogical
introduction to the analytic properties of the perturbative S-matrix from a modern point of
view. To help with the investigation of related questions we introduce a number of tools
including holomorphic cutting rules new approaches to dispersion relations as well as
formulae for local behavior of Feynman integrals near branch points. This 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.
