The Derivative Expansion of the Exact Renormalization Group

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. CERN-TH.7203/94 SHEP 93/94-16 hep-ph/9403340 March, 1994. ∗ On Leave from Southampton University, U.K. Our aim is to find a reliable and accurate analytic approximation method of general applicability to non-perturbative quantum field theory [2]–[4]. We believe that such a method must be based, in practice, on the Exact Renormalisation Group[1] (ERG), for reasons explained in ref.[2]. We particularly consider here the nonperturbative low energy behaviour – the Wilson fixed point[1] – of three dimensional one-component (Z2 invariant) scalar field theory, which is also the continuum limit of the three dimensional Ising model and physically the universality class of critical binary fluids and liquid-vapour transitions in classical fluids[11]. We choose to study this because of its relative simplicity but the method – just a derivative expansion of the Legendre effective action combined with the ERG – is certainly, appropriately modified[10], of much wider applicability to non-perturbative quantum field theory. In this letter we concentrate on developing a practical framework for such an expansion. The explicit calculations are only taken to second order, which is not sufficient (with our choice of cutoff) for our estimates of exponents to be competitive with the worlds best present estimates[11]; nevertheless, already the differences, between our results and those, are 6 or less times their quoted error, and our error appears to roughly halve with each new order of approximation. Finally we should emphasize that, in calculating directly the effective lagrangian, we may derive much more of physical interest than just critical exponents. The reader may well be asking why such a natural approximation scheme, built around ideas nearing twenty years old[1], has not been well studied before. It is as well to realise however that there are many contenders for methods of approximation, that sound perhaps equally natural, but do not work. It is also true that the computation must be organised carefully: merely expanding vertices in invariant polynomial combinations of momenta will soon grind to a halt through shear complexity. We list below methods that do not work in general. a) A sequence of truncations of Dyson-Schwinger equations, and other methods where renormalisability is a serious problem[2]. b) A sequence of truncations of the ERG, because these do not converge beyond a certain point, and because no completely reliable method could be found to reject 1 a.k.a. Wilson’s renormalisation group 2 For examples of other recent work on the ERG see refs.[5]–[10]. 1 the many spurious solutions that are also generated[3]–[5] (see however [6]). c) Direct numerical approaches to (truncations of) the ERG[2]. d) Approximations other than those that can be formulated for one particle irreducible parts of the vertices of the Wilson effective action[2]. e) Momentum (i.e. derivative) expansion of the ERG with smooth cutoff, if the effective width over which the cutoff varies is any less than the cutoff itself[2]. f) Momentum scale expansion of the ERG with sharp cutoff because only certain truncations are calculable in closed form[4] – but this is not good enough, c.f. (b). Because of (d) we use flow equations formulated directly for the Legendre effective action with respect to changes in some infrared cutoff scale Λ. This gives directly the one particle irreducible parts of the Wilson effective action as detailed in ref.[2]. Because of (c) we must approximate the momentum dependence and the only efficacious method we know of, is a momentum expansion[2]. Because of (e) and (f) we use a smooth cutoff with an effective width at least Λ (that is with little variation on scales less than Λ). This means that the cutoff will only somewhat suppress, and not eliminate, low energy modes. Finally because of (b) we perform only the derivative expansion and do not at all expand in the field φ. The partition function is defined as