Abstract
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 → Γ k 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 Γ Λ k and S k.
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Morris, T.R., Slade, Z.H. Solutions to the reconstruction problem in asymptotic safety. J. High Energ. Phys. 2015, 94 (2015). https://doi.org/10.1007/JHEP11(2015)094
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DOI: https://doi.org/10.1007/JHEP11(2015)094