Solutions to the reconstruction problem in asymptotic safety

Abstract

Abstract: Starting from a full renormalised trajectory for the effective average action (a.k.a. infrared cutoff Legendre effective action) Γk, we explicitly reconstruct corresponding bare actions, formulated in one of two ways. The first step is to construct the corresponding Wilsonian effective action Sk through a tree-level expansion in terms of the vertices provided by Γk. It forms a perfect bare action giving the same renormalised trajectory. A bare action with some ultraviolet cutoff scale Λ and infrared cutoff k necessarily produces an effective average action ΓkΛ that depends on both cutoffs, but if the already computed SΛ is used, we show how ΓkΛ can also be computed from Γk by a tree-level expansion, and that ΓkΛ → Γk as Λ → ∞. Along the way we show that Legendre effective actions with different UV cutoff profiles, but which correspond to the same Wilsonian effective action, are related through tree-level expansions. All these expansions follow from Legendre transform relationships that can be derived from the original one between ΓkΛ and Sk.