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...