Amplitudes

The increasing precision of experimental measurements at CERN's Large Hadron Collider (LHC) calls for increased precision in theoretical predictions. The key ingredients in these predictions are quantum scattering amplitudes. A new subfield of particle physics, called "Amplitudes", has emerged in recent years. It is devoted to the calculation, study, and application of scattering amplitudes. Its emergence was sparked by a new class of theoretical techniques, known collectively as "on-shell methods'', that made possible dramatic advances in calculations. Our proposal aims to develop a new generation of methods for computing physical quantities needed for the LHC experimental program. It will also create new links to related areas of research in theoretical physics and mathematics. The project will also train younger scientists, allowing them to acquire skills useful in broad areas of fundamental and applied research. We will develop methods directly addressing amplitudes in QCD theory; and we will also use maximally supersymmetric Yang-Mills theory as a laboratory for developing tools and probing new mathematical structures in quantum field theory. We have organized our goals into four primary objectives, which we will pursue through 12 tasks. Our first objective is to develop new methods for computing physical observables such as differential cross sections that avoid the conventional problem of infrared singularities in intermediate stages. We will pursue this objective through tasks devoted to computing energy-weighted cross sections beyond two loops in QCD and in maximally supersymmetric Yang-Mills; to computing them at finite coupling in the latter; and to relate these quantities in the two theories. Our second objective is to explore the analytic and algebraic structure of scattering amplitudes at finite coupling in the planar limit of the maximally supersymmetric Yang-Mills theory. To this end, we will exploit integrability, the AdS/CFT duality and the representation of amplitudes as light-like polygonal Wilson loops. We will pursue this objective through the study of Regge factorization of amplitudes at finite coupling; the calculation of amplitudes at strong coupling; and applying a recently-proposed triangulation technique to correlation functions and amplitudes. Our third objective is to elucidate surprising structures in amplitudes, such as the Bern-Carrasco-Johansson and Kawai-Lewellen-Tye relations. We will exploit string amplitudes for this purpose. We will pursue this objective through new derivations of kinematic relations between amplitudes; exposing the analytic structure of Feynman integrals and developing new means to evaluate them; and through eikonal calculations in Yang-Mills theory and in gravity. Our fourth objective is to develop new methods for the computation of amplitudes and cross sections required for high-precision studies at the LHC, applying insights from the other objectives to this purpose. We will pursue this objective through developments in the generalised-unitarity method for higher-loop amplitudes; new numerical approaches to evaluating Feynman integrals; and a new way of handling the intermediate infrared divergences in phase-space integration. The project follows the guiding spirit of many years of successful advances in Amplitudes: performing explicit calculations at the edge of what is feasible reveals new properties and structures; a deeper understanding of the uncovered features leads to new techniques, whose application makes possible new and even harder calculations, closing a virtuous circle.