The renormalisation group is a crucial tool for understanding scale-dependent quantum field theories. Renormalisation group fixed points correspond to theories where scale invariance is restored at the quantum level, and may provide high- or low-energy limits for more general quantum field theories. In particular, those reached in the ultraviolet allow theories to be defined microscopically, a scenario known as asymptotic safety. In this work I investigate fixed points of conventional four-dimensional, at-space, perturbatively renormalisable, local quantum field theories. Focusing on weakly interacting fixed points the problem becomes amenable to perturbation theory. The approach is twofold: on the one hand to understand general conditions for the existence of such fixed points, and on the other to construct theories which introduce new features compared to previous examples. To understand perturbative fixed points, general calculations for theories of this type are exploited. It is established, for gauge theories, interacting fixed points may be nonzero in gauge couplings alone, or in gauge and Yukawa couplings. Deriving novel group theory bounds it is established that only the latter may possibly be ultraviolet. Additionally it is shown that theories without gauge interactions cannot possess weakly coupled fixed points, and the connexion between this fact and the impossibility of such theories being asymptotically free is highlighted. Two explicit families of examples are presented: a theory with semisimple gauge group is analysed in detail, containing many new fixed points, a rich phase structure, and asymptotically safe regions of parameter space, and a separate supersymmetric model with an ultraviolet fixed point, providing the first known explicit example of an asymptotically safe supersymmetric gauge theory.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:767026 |
| Date | January 2018 |
| Creators | Bond, Andrew David |
| Publisher | University of Sussex |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://sro.sussex.ac.uk/id/eprint/80674/ |
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