Elements of the Continuous Renormalization Group

Abstract

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 <ver.0.8>

Aspects of the functional renormalisation group

“…The Functional Renormalisation Group (FRG) in its continuum formulation [1][2][3][4][5][6][7][8][9][10][11][12][13][14] has proven itself as a powerful tool for studying both perturbative and nonperturbative effects in quantum field theory and statistical physics, for reviews see [15][16][17][18][19][20][21][22][23]. In this approach a regularisation of a quantum theory is achieved by suppressing part of the propagating degrees of freedom related to a cut-off scale k. This results in regularised generating functionals such as the effective action Γ k where part of the modes have been integrated out.…”

On the renormalization of theories of a scalar chiral superfield

“…Moreover, the reparemtrization invariance of the flow equation is catastrophically broken. Indeed, as recognized by Wegner [41] and very nicely put by Morris [2], the ERG equation at a fixed point can be thought of as a non-linear eigenvalue equation for the anomalous dimension. So, the lowest order in the superderivative expansion looks to be useless for finding fixed points.…”

“…A selection of terms contributing to D (2) [or D (2) m ], by which we mean all D (i,j) with i + j = 2, is shown in figure 2.…”

“…It is very simple to show, nonperturbatively, that scale dependent renormalizable theories follow by considering flows out of some critical fixed point along the relevant and marginally relevant 2 directions as defined at this fixed point [2]. A crucial feature of these 1 Poles of this type can have different origins; those arising due to small/large loop momentum behaviour are known as renormalons-for a review see [4].…”

“…For theories which are perturbatively renormalizable but for which an interacting continuum limit based around the Gaussian fixed point nevertheless does not exist, ultraviolet (UV) renormalons give rise to the poles along the positive real axis. 2 Henceforth, we will take 'relevant' to include marginally relevant.…”