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The c and a-theorems and the Local Renormalisation Group

Graham Shore

SpringerBriefs in Physics

Swansea University Author: Graham Shore

DOI (Published version): 10.1007/978-3-319-54000-9

Abstract

The Zamolodchikov c-theorem has led to important new insights in the understanding of the Renormalisation Group (RG) and the geometry of the space of QFTs. The present primer introduces and reviews the parallel developments of the search for a higher-dimensional generalisation of the c-theorem and o...

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Published in: SpringerBriefs in Physics
ISBN: 9783319539997 9783319540009
Published: Cham, Switzerland Springer International Publishing 2017
Online Access: http://www.springer.com/gp/book/9783319539997
URI: https://cronfa.swan.ac.uk/Record/cronfa26479
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Abstract: The Zamolodchikov c-theorem has led to important new insights in the understanding of the Renormalisation Group (RG) and the geometry of the space of QFTs. The present primer introduces and reviews the parallel developments of the search for a higher-dimensional generalisation of the c-theorem and of the Local RG (LRG).The idea of renormalisation with position-dependent couplings, running under local Weyl scaling, is traced from its early realisations to the elegant modern formalism of the LRG. The key rôle of the associated Weyl consistency conditions in establishing RG flow equations for the coefficients of the trace anomaly in curved spacetime, and their relation to the c-theorem and four-dimensional a-theorem, is explained in detail.A number of different derivations of the c-theorem in two dimensions are presented and subsequently generalised to four dimensions. The obstructions to establishing monotonic C-functions related to the trace anomaly coefficients in four dimensions are explained. The possibility of deriving an a-theorem for the coefficient of the Euler-Gauss-Bonnet density is explored, initially by formulating the QFT on maximally symmetric spaces. Then the formulation of the weak a-theorem using a dispersion relation for four-point functions is presented.Finally, the application of the LRG to the issue of limit cycles in theories with a global symmetry is described, shedding new light on the geometry of the space of couplings in QFT.
Item Description: arXiv version: https://arxiv.org/abs/1601.06662
Keywords: Mathematical physics, Quantum Field Theories, String Theory, Mathematical Physics, Mathematical Methods in Physics, Physics, Renormalization group, SCIENCE Energy, Mechanics General, Physics, General, Science Quantum Theory, Local Weyl scaling, Position-dependent couplings in QFT, Zamolodchikov c-theorem, Global symmetry and limit cycles, Weyl consistency condition.
College: Faculty of Science and Engineering