Momentum scale expansion of sharp cutoff flow equations

Abstract

We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organised by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor expansions not being possible). A systematic series of approximations -the O(p M ) approximations -result from discarding from these parts, all terms of higher than the M th degree. These approximations preserve a field reparametrization invariance, ensuring that the field's anomalous dimension is unambiguously determined. The lowest order approximation coincides with the local potential approximation to the Wegner-Houghton equations. We discuss the practical difficulties with extending the approximation beyond O(p 0 ). hep-th/9508017August, 1995. via the one-particle irreducible parts, and the best that one can achieve is an expansion in momentum scale[13] p ∼ √ p µ p µ , the M th approximation resulting from dropping all terms beyond O(p M ). This raises the spectre that the lowest order approximation -O(p 0 ), in which the momentum scale dependence is completely discarded, nevertheless still involvesGreen functions with non-trivial dependence on the angles between their momenta. It turns out that this dependence cancels however, and it is in this way that we recover the local potential approximation [4][5][6][7][8][9][10][11][12].Our second main motivation is unfortunately more negative. We wish to provide the evidence for the claim in ref. [14]: that the momentum scale expansion of the sharp cutoff flow equations is not a practical technique, i.e. beyond O(p 0 ), because only certain truncations (of the field dependence) are calculable in closed form, and this is not good enough[8]. It is not good enough because, in contrast to a momentum scale expansionwhich might be expected to work 5 since it corresponds to an expansion in (an appropriately defined) 'localness' of the effective Lagrangian, truncations of the field dependence have limited accuracy and reliability[8]. We want to emphasise here that our claim that there are practical difficulties in extending the momentum scale expansion beyond O(p 0 ), merely amounts to an admission of our own lack of ability in extending it in a substantive way. It would certainly be worthwhile to improve on these attempts (presented in sect.6), because a simple two-loop example -and general arguments (c.f. ref.[13] and appendix B), suggest that the sharp cutoff momentum scale expansion should yield the fastest convergence (i.e.faster than the smooth cutoff momentum expansions [14,15]).Despite the increasing interest in utilising the exact renormalization group to develop non-perturbative approximations [19], there have so far been relatively few works concerned with going beyond the local potential approximation [11][12][13][14][15][16][17][18]. Part of the point of sect.6 is also pedagogical: we provide simple concrete illustrations of the discussions in earlier sections, but also reemphasise [11,[14][15][16]20] in a simple context the importan...

Fate of nonpolynomial interactions in scalar field theory

“…But the same equation can also be reached by taking the sharp cutoff limit of any exact renormalisation group equation [67,75].…”

“…Halpern and Huang noticed that, within the (LPA) Local Potential Approximation [69][70][71][72][73][74][75][76][77] linearised perturbations about the Gaussian fixed point not only have solutions that are polynomial in the scalar field φ, which can be identified with the operators usually considered in perturbation theory, but also solutions that are non-polynomial in the field with a continuous scaling dimension [58,60]. 5 It is a particular set of these latter perturbations that Halpern and Huang identified as being asymptotically free.…”

Exact Renormalization Group Approach in Scalar and Fermionic Theories

“…In the momentum representation Eq. (22) corresponds to the expansion in powers of momenta, so one can hope that using a few first terms of this expansion is justified if we consider effects at low momenta (there may be some complications in the case of the sharp cutoff 21 ). The function V (φ, t) is the effective (local) potential of the theory.…”

“…(27) are basically β-functions for the coupling constants of corresponding operators. The leading order of the derivative expansion with subsequent polynomial approximation of the potential V (φ, t) in scalar theories for the Wegner-Houghton ERG equation was studied in detail 27,28,21 . We summarize some of the results in the next section.…”

Critical behavior of weakly interacting bosons: A functional renormalization-group approach

“…Problems with a similar implementation of the sharp cutoff formulation were previously reported in Ref. [26]. Ignoring the third order terms in the flow of v l , Eq.…”