And the Exact Renormalization Group *

We compute critical exponents in a Z2 symmetric scalar field theory in three dimensions, using Wilson’s exact renormalization group equations expanded in powers of derivatives. A nontrivial relation between these exponents is confirmed explicitly at the first two orders in the derivative expansion. At leading order all our results are cutoff independent, while at next-to-leading order they are not, and the determination of critical exponents becomes ambiguous. We discuss the possible ways in which this scheme ambiguity might be resolved. November 1994 ∗ This work is supported in part by funds provided by Grant AEN 90-0033 (Spain), and by the M.E.C. (Spain). † On leave from a Royal Society University Research Fellowship. a E-mail: rball@surya11.cern.ch b E-mail: haagense@cinelli.physics.mcgill.ca c E-mail: latorre@ecm.ub.es, moreno@ecm.ub.es The exact renormalization group [1-4] is in many instances a powerful analytical technique for the study of nonperturbative renormalization group flows. In practice one must always truncate an infinite tower of equations, and in the past surprisingly good results for critical exponents and renormalization group flows of scalar field theories in 2 ≤ d ≤ 4 have been obtained [5-10] by suppressing all the equations except that for the effective potential. There have been several attempts [11-14] to improve the accuracy of these calculations by also keeping terms in the effective action containing two derivatives of the fields, and thus allowing for wave function renormalization. In this letter we will consider the important issue of scheme dependence within this context. A Euclidean field theory may in general be regularized by employing a certain cutoff function KΛ(p), which suppresses modes with momenta higher than some scale Λ. The exact renormalization group equations are then constructed so as to ensure that as Λ is reduced, and thus more and more modes are integrated out, the vertex functions of the theory are varied in such a way that the Green’s functions remain unchanged. This provides a nonperturbative definition of the theory. Now in principle physical observables should be determined unambiguously for a broad class of cutoff functions KΛ(p). This is what we mean by scheme independence. Indeed in certain truncation schemes this independence can be proven; for instance, it can be shown that S-matrix elements are scheme independent to any given order in perturbation theory[15], even though beta-functions and anomalous dimensions are in general scheme dependent beyond leading order. However in other truncations scheme independence cannot be taken for granted. In the derivative expansion, in particular, this problem has not so far been considered: a particular choice of cutoff function is made (e.g., sharp cutoffs [5-9], power-like cutoffs [13], or exponential cutoffs [12,14]), giving perhaps the impression that the results obtained are in some sense unique. Here we will compute various critical exponents for a Z2-symmetric scalar field theory in d=3 dimensions using the simple and elegant exact renormalization group equations due to Wilson [1] and Polchinski [4]: previous authors [5-9,13,14] have based their computations on the Wegner-Houghton [2] and Weinberg [3] equations. We work to all orders in the fields, but expand the (Wilson) effective action in powers of derivatives. We will show that all the scheme dependence can be absorbed into 2n parameters at n-th order in the expansion. At leading order, n = 0, the critical exponents are then scheme independent: they are independent of the particular choice of cutoff function. However at next-to-leading order the results become scheme dependent. This makes it particularly difficult to compute