GeoScatAmp · The Geometry of Scattering Amplitudes

The Geometry of Scattering Amplitudes

Perturbative quantum field theory is the basis of much of the modern description of fundamental interactions. Its natural observables are cross-sections in scattering experiments, and these cross-sections are computed from the quantum-mechanical probability amplitude for a given outcome of the scattering. Over the past decade, inspired by the possibilities of the LHC, the theory community has made tremendous progress in our understanding of scattering amplitudes. In the effort to sidestep the difficulties of field theory calculations, surprising new structures and symmetries have been uncovered, which are hidden in the traditional Feynman diagram approach to quantum field theory. During my fellowship, I have investigated these new approaches, exploring their applicability in a variety of scenarios, also beyond the topic of scattering amplitudes. I am very happy with the progress of my research. The major goals of the proposal were achieved, and the results have opened up new and important lines of work. This project led to several articles published in some of the most respected journals in my field (see Publications list), and to several invitations for me and my collaborators to present our work at departmental seminars and international conferences (see Dissemination Activities list). I can divide the main results into three topics, which are roughly in chronological order. 1. Understanding the kinematic algebra of gauge theory and gravity amplitudes: Gauge theory (a generalisation of the theory of electromagnetism) and gravity are the two most fundamental interactions describing Nature. There are surprising connections between the scattering amplitudes of these theories, which fall under the slogan “gravity = gauge theory squared”. I have investigated the mathematical structure underlying these connections, finding that it is based on a certain kinematic algebra, known from my previous work for a specific sector of these theories. In [4] (the numbers here and below refer to the Publications list), we understood how to extend this algebra to any sector of the theories, in the classical limit. The main tool was the appearance of the so-called scattering equations, which underlie beautiful new formulas for the scattering amplitudes of several theories. In a more recent publication (now submitted to a journal), we extend the previous work in a different direction, finding a highly non-trivial expression of “gravity = gauge theory squared” for a class of one-loop amplitudes, the first quantum correction to the classical result. 2. Beyond scattering amplitudes: The new mathematical structures found in quantum field theory have an impact beyond the realm of scattering amplitudes. It has been one of my goals to explore these structures more generally. In two articles [2,5] (see also [6]), we have applied relations of the type “gravity = gauge theory squared” to exact classical solutions, finding a strikingly simple connection between well-known gravity solutions, such as the Schwarzschild black hole, and gauge theory solutions, such as the Coulomb potential. While there was an earlier intuition that these historically important solutions should be related, our results provide the exact statement of that relation, and extend it to an infinite class of examples. One of the great surprises is that the physics of scattering amplitudes is so closely related to the physics of exact classical solutions. 3. Scattering equations and worldsheet models: My work on the kinematic algebra led me to the study of the scattering equations, as mentioned in point 1. These equations have led to surprising formulas for the scattering amplitudes of certain quantum field theories, which suggest a connection to string theory. Indeed, it has been found that these formulas are related to the existence of worldsheet models inspired by string theory. In worldsheet models, interactions are not point-like, as in ordinary quantum field theory, but are instead encoded in a two-dimensional surface called the worldsheet. In [3], prompted by the formulas for amplitudes based on the scattering equations, we constructed worldsheet models for several quantum field theories of interest. More important, however, was our extension of the scattering equations to loop-level amplitudes, i.e. the quantum corrections, which are notoriously hard to calculate. We showed that the use of the scattering equations at loop level is not much harder that the leading (classical) contribution to the amplitude. This is a result which can be expected to have a widespread impact in the field. I thank the Marie Curie Actions for the support I received over the past two years. Note: This report was written by the Fellow and approved by the Supervisor.

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