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) $Γ_k$, 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 $Γ_k$. 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 $Γ^Λ_k$ that depends on both cutoffs, but if the already computed $S^Λ$ is used, we show how $Γ^Λ_k$ can also be computed from $Γ_k$ by a tree-level expansion, and that $Γ^Λ_k\toΓ_k$ as $Λ\to\infty$. 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 $Γ^Λ_k$ and $S^k$.

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