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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 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 fix 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 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 Feynmanintegrals 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.

Using the analyticity property of the amplitudes of physical processes allows one to obtain important relations connecting various physical quantities. These relations can be established without specifying the concrete form of the interaction between particles, which is often poorly known. The analytic methods make use of the unitarity property of the S-matrix and the formalism of the Feynman diagrams. The present book is devoted to a systematic description of the analytic and diagram methods in the theory of nuclear reactions at low and intermediate energies. First, a general overview of the unitarity and analyticity properties of the S- and T-matrices is presented. Then the general theory of non-relativistic Feynman diagrams, their specific features and the analytic properties of their amplitudes is described. Three- and four-prong vertex functions of the non-relativistic Feynman diagrams are among the topics that are addressed and discussed, including the important relation between vertex functions and the asymptotic form of the nuclear bound-state wave functions. Specific approaches to describing nuclear reactions, which use analytic and diagram methods are outlined. These methods include the dispersion K-matrix approach, the dispersion relations, and the Trojan horse method. A part of the book deals with the asymptotic normalization coefficients (ANCs). ANCs are fundamental nuclear characteristics that are important in nuclear reaction and nuclear structure physics as well as in nuclear astrophysics. They are used actively in the analysis of nuclear reactions, in particular in considering the creation of elements in the Universe. Various methods of determining ANCs are overviewed. One chapter of the book is devoted to the methods of obtaining information on the features of bound nuclear states by an analytic continuation of the scattering data. The analytic continuation of the Lippmann-Schwinger and Faddeev integral equations onto unphysical energy sheets, which can be used for finding the position and width of resonances and the virtual (antibound) states is described. A generalization of wave function normalization to the Gamow states is outlined.