Solutions to the reconstruction problem in asymptotic safety

Starting from a full renormalised trajectory for the effective average action (a.k.a. infrared cutoff Legendre effective action) Gamma(k), we explicitly reconstruct corresponding bare actions, formulated in one of two ways. The first step is to construct the corresponding Wilsonian effective action S-k through a tree-level expansion in terms of the vertices provided by Gamma(k). It forms a perfect bare action giving the same renormalised trajectory. A bare action with some ultraviolet cutoff scale Lambda and infrared cutoff k necessarily produces an effective average action Gamma(Lambda)(k) that depends on both cutoffs, but if the already computed S-Lambda is used, we show how Gamma(Lambda)(k) can also be computed from Gamma(k) by a tree-level expansion, and that Gamma(Lambda)(k) -> Gamma(k) as Lambda -> infinity. 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 Gamma(Lambda)(k) and S-k.