The Renormalization Group and Two Dimensional Multicritical Effective Scalar Field Theory

Direct verification of the existence of an infinite set of multicritical non-perturbative FPs (Fixed Points) for a single scalar field in two dimensions, is in practice well outside the capabilities of the present standard approximate non-perturbative methods. We apply a derivative expansion of the exact RG (Renormalization Group) equations in a form which allows the corresponding FP equations to appear as non-linear eigenvalue equations for the anomalous scaling dimension $\eta$. At zeroth order, only continuum limits based on critical sine-Gordon models, are accessible. At second order in derivatives, we perform a general search over all $\eta\ge.02$, finding the expected first ten FPs, and {\sl only} these. For each of these we verify the correct relevant qualitative behaviour, and compute critical exponents, and the dimensions of up to the first ten lowest dimension operators. Depending on the quantity, our lowest order approximate description agrees with CFT (Conformal Field Theory) with an accuracy between 0.2\% and 33\%; this requires however that certain irrelevant operators that are total derivatives in the CFT are associated with ones that are not total derivatives in the scalar field theory.Comment: Note added on "shadow operators". Version to be published in Phys. Lett.

Renormalization group properties of the conformal sector: towards perturbatively renormalizable quantum gravity

The Wilsonian renormalization group (RG) requires Euclidean signature. The conformal factor of the metric then has a wrong-sign kinetic term, which has a profound effect on its RG properties. Generically for the conformal sector, complete flows exist only in the reverse direction (i.e. from the infrared to the ultraviolet). The Gaussian fixed point supports infinite sequences of composite eigenoperators of increasing infrared relevancy (increasingly negative mass dimension), which are orthonormal and complete for bare interactions that are square integrable under the appropriate measure. These eigenoperators are non-perturbative in $\hbar$ and evanescent. For $\mathbb{R}^4$ spacetime, each renormalised physical operator exists but only has support at vanishing field amplitude. In the generic case of infinitely many non-vanishing couplings, if a complete RG flow exists, it is characterised in the infrared by a scale $\Lambda_\mathrm{p}>0$, beyond which the field amplitude is exponentially suppressed. On other spacetimes, of length scale $L$, the flow ceases to exist once a certain universal measure of inhomogeneity exceeds $O(1)+2\pi L^2\Lambda^2_\mathrm{p}$. Importantly for cosmology, the minimum size of the universe is thus tied to the degree of inhomogeneity, with spacetimes of vanishing size being required to be almost homogeneous. We initiate a study of this exotic quantum field theory at the interacting level, and discuss what the full theory of quantum gravity should look like, one which must thus be perturbatively renormalizable in Newton's constant but non-perturbative in $\hbar$.Comment: 52 pages, 4 figures; fixed typos; improved explanation of the sign of V, and the use of Sturm-Liouville theory. To be publ in JHE

Elements of the Continuous Renormalization Group

These two lectures cover some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will describe how real non-perturbative calculations can be done, particularly within derivative expansion approximations. An effort has been made to keep the lectures pedagogical and self-contained. Topics covered are the derivation of the flow equations, their equivalence, continuum limits, perturbation theory, truncations, derivative expansions, identification of fixed points and eigenoperators, and the role of reparametrization invariance. Some new material is included, in particular a demonstration of non-perturbative renormalizability, and a discussion of ultraviolet renormalons.Comment: Invited lectures at the Yukawa International Seminar '97. 20 pages including 6 eps figs. LaTeX. PTPTeX style files include

Convergence of derivative expansions of the renormalization group

We investigate the convergence of the derivative expansion of the exact renormalization group, by using it to compute the beta function of scalar field theory. We show that the derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. The derivative expansion of the Legendre flow equation trivially converges at one loop, but also at two loops: slowly with sharp cutoff (as a momentum-scale expansion), and rapidly in the case of a smooth exponential cutoff. Finally, we show that the two loop contributions to certain higher derivative operators (not involved in beta) have divergent momentum-scale expansions for sharp cutoff, but the smooth exponential cutoff gives convergent derivative expansions for all such operators with any number of derivatives.Comment: Latex inc axodraw. 20 page

Redundant operators in the exact renormalisation group and in the f(R) approximation to asymptotic safety

In this paper we review the definition and properties of redundant operators in the exact renormalisation group. We explain why it is important to require them to be eigenoperators and why generically they appear only as a consequence of symmetries of the particular choice of renormalisation group equations. This clarifies when Newton’s constant and or the cosmological constant can be considered inessential. We then apply these ideas to the Local Potential Approximation and approximations of a similar spirit such as the f (R) approximation in the asymptotic safety programme in quantum gravity. We show that these approximations can break down if the fixed point does not support a ‘vacuum’ solution in the appropriate domain: all eigenoperators become redundant and the physical space of perturbations collapses to a point. We show that this is the case for the recently discovered lines of fixed points in the f (R) flow equations