The numerically optimized regulator and the functional renormalization group

Abstract

We aim to optimize the functional form of the compactly supported smooth (CSS) regulator within the functional renormalization group (RG), in the framework of bosonized two-dimensional quantum electrodynamics (QED2) and of the three-dimensional O(N = 1) scalar field theory in the local potential approximation (LPA). The principle of minimal sensitivity (PMS) is used for the optimization of the CSS regulator, recovering all the major types of regulators in appropriate limits. Within the investigated class of functional forms, a thorough investigation of the CSS regulator, optimized with two different normalizations within the PMS method, confirms that the functional form of a regulator first proposed by Litim is optimal within the LPA. However, Litim’s exact form leads to a kink in the regulator function. A form of the CSS regulator, numerically close to Litim’s limit while maintaining infinite differentiability, remains compatible with the gradient expansion to all orders. A smooth analytic behavior of the regulator is ensured by a small, but finite value of the exponential fall-off parameter in the CSS regulator. Consequently, a compactly supported regulator, in a parameter regime close to Litim’s optimized form, but regularized with an exponential factor, appears to have favorable properties and could be used to address the scheme dependence of the functional RG, at least within the approximations employed in the studies reported here.