Analytic Structure and Finiteness of Scattering Amplitudes

Abstract

Scattering amplitudes are fundamental objects in high energy physics, providing a bridge between theoretical calculations and data from particle colliders. The framework underpinning the study of elementary particles - quantum field theory - has led to remarkably precise scattering amplitude predictions, which, in turn, provide a foundation for fundamental physics discoveries. Despite this success, particles remain poorly described within this framework. Consequently, in theories with massless particles, asymptotic interactions render scattering amplitudes infrared divergent in perturbation theory and ill-defined non-perturbatively. In this dissertation, we take steps towards strengthening the theoretical foundations of quantum field theory, by defining infrared finite cross sections and amplitudes, and by probing the analytic structure of amplitudes through examining their sequential discontinuities. First, we show that infrared finite cross sections are obtained by summing over either initial or final states, and explore how forward scattering diagrams often constitute a crucial contribution to achieve finiteness. Then, using universality and factorization of asymptotic interactions, we demonstrate how to define finite scattering amplitudes in gauge theories. Exploiting freedom in choosing regulators and cutoffs for the asymptotic interactions, these amplitudes can be interpreted alternatively as Wilson coefficients, as remainder functions, or as coherent states. Finally, we extend the traditional cutting rules, that relate discontinuities of amplitudes to cuts of the corresponding Feynman diagrams, to sequential discontinuities and multiple cuts. Our relations provide a new probe of the analytic properties of amplitudes, in addition to a scheme for exploring the finite scattering amplitudes further. Through an enhanced understanding of the analytic structure of finite amplitudes, we hope to unveil a proper description of particles in quantum field theory.