We investigate the structure of Polchinski's formulation of the flow equations for the continuum Wilson effective action. Reinterpretations in terms of I.R. cutoff greens functions are given. A promising non-perturbative approximation scheme is derived by carefully taking the sharp cutoff limit and expanding in 'irrelevancy' of operators. We illustrate with two simple models of four dimensional λϕ 4 theory: the cactus approximation, and a model incorporating the first irrelevant correction to the renormalized coupling. The qualitative and quantitative behaviour give confidence in a fuller use of this method for obtaining accurate results.

The functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximated by a derivative expansion; no other approximation is made. This results in a set of coupled non-linear differential equations. The corresponding differential equations for a fixed point action have at most a countable number of solutions that are well defined for all values of the field. We apply the technique to the fixed points of one-component real scalar field theory in three dimensions. Only two non-singular solutions are found: the gaussian fixed point and an approximation to the Wilson fixed point. The latter is used to compute critical exponents, by carrying the approximation to second order. The results appear to converge rapidly.

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 rôle of reparametrization invariance. Some new material is included, in particular a demonstration of non-perturbative renormalisability, and a discussion of ultraviolet renormalons. §1. IntroductionAs stated above, these lectures will concentrate on exact statements, the conceptual advances, in the exact renormalization group, a.k.a. Wilson's continuous renormalization group: 1) This is motivated by the belief that these are ultimately the most important aspects of the recent progress, but at the same time this viewpoint lends itself to a (hopefully) elegant and pedagogical introduction to this area. This means however that applications will not be reviewed, or practical matters such as the accuracy of approximations discussed per se. Individuals interested to learn more about these issues, are encouraged to consult our reviews 2), 3) and the lectures by Aoki and Wetterich in this volume. Suffice to say here that there are approximations, in particular the derivative expansion, which give fair to accurate numerical results in practice. The motivation fueling the recent progress is the need to derive better analytic approximation methods for truly non-perturbative quantum field theory, i.e. where there are no small parameters * ) one can fruitfully expand in. There is a clear need for such approaches, of course within the archetypical example -low energy QCD, but perhaps more importantly in the need to better understand (even qualitatively) the possibilities offered by the full parameter space of non-perturbative quantum field theories, such as may explain some of the mysteries of the symmetry breaking sector of the Standard Model (for example), and/or Planck scale physics. On the other hand, the issue of renormalisability, which in many approaches ap- * ) Typical small parameters that are sometimes useful are small coupling, i.e. perturbation theory, or 1/N where N is the number of components of a field, or = 4 − D where D is the space-time dimension.typeset using PTPT E X.sty

We investigate the Exact Renormalization Group (ERG) description of (Z 2 invariant) one-component scalar field theory, in the approximation in which all momentum dependence is discarded in the effective vertices. In this context we show how one can perform a systematic search for non-perturbative continuum limits without making any assumption about the form of the lagrangian. The approximation is seen to be a good one, both qualitatively and quantitatively. We then consider the further approximation of truncating the lagrangian to polynomial in the field dependence. Concentrating on the non-perturbative three dimensional Wilson fixed point, we show that the sequence of truncations n = 2, 3, . . ., obtained by expanding about the field ϕ = 0 and discarding all powers ϕ 2n+2 and higher, yields solutions that at first converge to the answer obtained without truncation, but then cease to further converge beyond a certain point. Within the sequence of truncations, no completely reliable method exists to reject the many spurious solutions that are also generated. These properties are explained in terms of the analytic behaviour of the untruncated solutions -which we describe in some detail.

Abstract:We uncover a method of calculation that proceeds at every step without fixing the gauge or specifying details of the regularisation scheme. Results are obtained by iterated use of integration by parts and gauge invariance identities. The initial stages can even be computed diagrammatically. The method is formulated within the framework of an exact renormalization group for SU (N ) Yang-Mills gauge theory, incorporating an effective cutoff through a manifest spontaneously broken SU (N |N ) gauge invariance. We demonstrate the technique with a compact calculation of the one-loop beta function, achieving a manifestly universal result, and without gauge fixing, for the first time at finite N .