Aspects of the Renormalisation Group for gauge theories and gravity

Abstract

This thesis is devoted to exploring various aspects of the renormalisation group for gauge theories and gravity. After introducing the necessary concepts, we study a manifestly gauge invariant and background independent ERG flow equation for gravity and Yang- Mills theories, which seeks to bypass the usual problems regarding gauge-fixing and ghosts by implementing a geometric version of the higher covariant derivative regularisation scheme. In the process we develop the machinery needed to study it rigorously and show that, under rather loose assumptions, the regularisation is at odds with gauge invariance. Next we investigate scalar field theories within the Local Potential Approximation (LPA). We show that in the large field limit the eigenoperator equation can be treated as a Sturm-Liouville (SL) problem, which in turn allows us to recast it as a Schrodinger type equation, and compute the scaling dimension of the corresponding eigenoperators using Wentzel-Kramers-Brillouin (WKB) analysis. We find that the scaling dimension for the O(N ) case is twice that for a single scalar field, and moreover that both results are universal and independent of the details of the regularisation used. Finally, we consider off-shell perturbative renormalisation of pure quantum gravity. We show that at each new loop order the divergences that do not vanish on-shell are con- structed from only the total metric, whilst those that vanish on-shell are renormalised by canonical transformations involving the quantum fields. Purely background metric divergences do not separately appear and the background metric does not get renormalised. We verify these assertions by computing leading off-shell divergences to two loops, exploiting off-shell BRST invariance and the renormalisation group equations. Furthermore, although some divergences can be absorbed by field redefinitions, we ex- plain why this does not lead to finite beta functions for the corresponding field.

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Identifiers

ORCID for Vlad Mihai Mandric: orcid.org/0000-0003-2771-0945

ORCID for Tim Morris: orcid.org/0000-0001-6256-9962

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