Convergence and Stability of the Renormalisation Group

Renormalisation group techniques are important tools to describe how classical physics is modified by quantum fluctuations. Integrating-out all quantum fluctuations provides the link between the classical theory and the full quantum effective theory [1]. A useful method is given by the Exact Renormalisation Group (ERG) [2], which is based on the Wilsonian idea of integratingout infinitesimal momentum shells. ERG flows have a simple one-loop structure. They admit non-perturbative truncations and are not bound to weak coupling. An application of the ERG requires some approximations like the derivative expansion or expansions in powers of the fields. It has been known since long that approximations induce a spurious dependence on the regularisation [3, 4, 5, 6, 7]. This is somewhat similar to the scheme dependence within perturbative QCD, or within truncated solutions of Schwinger-Dyson equations. While this scheme dependence should vanish at sufficiently high order in the expansion, practical applications are always bound to a finite order, and hence to a non-vanishing scheme dependence. A partial understanding of the interplay of approximations and scheme dependence has been achieved previously. For scalar QED [8], the scheme dependence in the region of first order phase transition has been studied in [4, 5]. For 3d scalar theories, the interplay between the smoothness of the regulator and the resulting critical exponents has been addressed in [9] using a minimum sensitivity condition. The weak scheme dependence found in these cases suggests that higher order corrections remain small, thereby strengthening the results existing so far. In this contribution, we review how the convergence and stability of ERG flows is optimised, thereby providing improved results already to low orders within a given approximation [10, 11, 12, 13, 14]. This involves a discussion on the origin of the spurious scheme dependence, and its link with convergence and stability properties of truncated ERG flows. We exemplify the basic reasoning for the universality class of O(N) symmetric scalar theories in three dimensions. It is expected that insights gained from this investigation will also prove useful for applications to more complex scalar theories, gauge theories [15] or gravity [16], which are more difficult to handle. 1Invited talk presented at RG2002, March 10-16, 2002, Strba, Slovakia. E-mail address: [email protected]