Description
- Abstract:
- Scattering amplitudes have undergone considerable study in the last few years. Much of the progress has come from abandoning Feynman diagram techniques and instead exploring and exploiting the physical constraints and mathematical structures underlying amplitudes. In this dissertation we present a new, unexpected structure underlying certain amplitudes: cluster algebras. Harnessing the power of cluster algebras allows us to calculate previously unknown amplitudes and points the way towards a deeper mathematical understanding of quantum field theory. We begin by introducing motivic amplitudes, which contain all of the essential mathematical content of scattering amplitudes in planar N=4 supersymmetric Yang-Mills theory. We then establish, through explicit calculations of the two-loop, seven-particle motivic amplitude as well as the n-particle two-loop differential, that the amplitude only depends on on certain preferred coordinates known in the mathematics literature as cluster X-coordinates on Conf_n(P^3). The connection between scattering amplitudes and cluster algebras prompts us to define cluster polylogarithm functions, objects which elegantly (and uniquely) contain beautiful motivic and cluster algebraic structure. In particular, cluster polylogarithms allow us to associate specific polylogarithm functions to faces of generalized associahedrons, to which cluster algebras have a natural combinatoric connection. These functions form a sufficient basis to express two-loop amplitudes, and we present an analytic formula for the two-loop seven-particle amplitude as an example. Furthermore, we find intriguing connections between motivic amplitudes and the geometry of associahedrons. For example, we show that the obstruction to the two-loop motivic amplitude being expressible in terms of classical polylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron.
- Notes:
- Thesis (Ph.D. -- Brown University (2015)
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Citation
Golden, John Kimbell,
"Cluster Polylogarithms and Scattering Amplitudes"
(2015).
Physics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0668BJB
