Solutions to the reconstruction problem in asymptotic safety

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 .

The unitary conformal field theory behind 2D Asymptotic Safety

“…The renormalization group flow has a non-Gaussian fixed point with a Newton constant of order ε; a result which is independent of the underlying regularization scheme and which is found in both perturbative and nonperturbative investigations. Employing a reconstruction formula [9,12,29] we shall see that this property holds not only for the effective, but also for the bare action: using an appropriate regularization prescription the bare Newton constant is of first order in ε, too. This is our motivation for considering a generic Einstein-Hilbert action with a Newton constant proportional to ε.…”

“…While this procedure is standard and familiar from perturbative quantum gravity and Yang-Mills theory, for instance, the situation is much more involved in Asymptotic Safety. The reason is that, implicitly, this indefinite metric quantization is applied to a bare action which is essentially given by the fixed point functional [9][10][11][12]. As such it is already in itself the result of a technically hard nonperturbative computation which in practice can be done only approximately, for the time being.…”

“…By construction, Γ k approaches the full quantum effective action Γ in the limit k → 0. Although its k → ∞ limit is formally related to the bare action [15][16][17], in the Asymptotic Safety program no bare action is posited a priori; it is rather "reconstructed" from the condition that this limit actually exists nonperturbatively [9][10][11][12].…”

“…Then S ind is proportional to the central charge c S , 12) and by (4.9) the action S ind gives rise to a stress tensor whose trace is precisely of the form (4.10). (The dots represent a cosmological constant term.)…”

“…Therefore, if in 2D the necessity of gauge fixing arises, we can in principle pick a new gauge, different from the one employed in d > 2 for the computation of the β-functions. 12 11 Which might not be realized in more complicated truncations! 12 This could not be done if one wants to combine loop or RG calculations from d > 2 with others done in d = 2 exactly.…”