The c and a-Theorems and the Local Renormalisation Group

Abstract

Abstract:The Zamolodchikov c-theorem has led to important new insights in our understanding of the renormalisation group and the geometry of the space of QFTs. Here, we review the parallel developments of the search for a higher-dimensional generalisation of the c-theorem and of the Local Renormalisation Group.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 local renormalisation group. 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 -using spectral functions, RG analysis of Green functions of the energy-momentum tensor T µν , and dispersion relations -and are generalised to four dimensions. The obstruction to establishing monotonic C-functions related to the β c and β b trace anomaly coefficients in four dimensions is discussed. The possibility of deriving an a-theorem, involving the coefficient β a of the Euler-Gauss-Bonnet density in the trace anomaly, 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 of T µ µ is presented. Finally, we describe the application of the local renormalisation group to the issue of limit cycles in theories with a global symmetry and it is shown how this sheds new light on the geometry of the space of couplings in QFT.

Constraints on 3- and 4-loop β-functions in a general four-dimensional Quantum Field Theory

“…thus establishing the perturbative a-theorem for leading-order positive-definite G IJ . For RG flows that include limit cycles, the interpretation of this is that G IJ has the potential to act as a metric on the quotient space M = g I / G K , where couplings related by G K transformations are identified, such that B I = 0 define fixed points on M, and the associated QFT is conformal [5,19]. A final extension of the LRG formalism includes the introduction of dimension-2 operators.…”

“…All TSs that do not correspond to 1PI vertex or field-strength graphs are eliminated, as they do not appear in MS-like schemes. The remaining TSs are thinned to a basis using gauge identities and each is given a coefficient 19 . At sufficiently-high loop order, the shift β → B is taken into account by identifying any possible antisymmetric combinations of 2-point TSs, attaching to a leg, and assigning a new arbitrary coefficient.…”

“…Clearly, equations (5.9a), (5.9b), (5.9e), (5.9f), (5.9g) and (5.9h) are completely new, as they do not appear in the linear combinations. The final piece of information fixes an underdetermination involving equations X i , i ∈ {10, 11,12,13,15,17,19,20, 23}: attempting to solve X i in terms of Y i leaves one free parameter, hence knowing any one of these nine conditions determines the rest.…”

On Newton-Cartan local renormalization group and anomalies

“…Positivity of Zamolodchikov metric, for unitary theories, in turn follows from the positivity of the spectral density function in the Källén-Lehmann spectral representation, see [20] for a review.…”

“…It is a very useful tool to study relativistic RG flows nearby conformal fixed points in various dimensions [11][12][13][14][15][16] and it has also interesting applications in the supersymmetric case [13,17,18]. For reviews see [19,20].…”