Equivalence of Local Potential Approximations

In recent papers it has been noted that the local potential approximation of the Legendre and Wilson-Polchinski flow equations give, within numerical error, identical results for a range of exponents and Wilson-Fisher fixed points in three dimensions, providing a certain “optimised” cutoff is used for the Legendre flow equation. Here we point out that this is a consequence of an exact map between the two equations, which is nothing other than the exact reduction of the functional map that exists between the two exact renormalization groups. We note also that the optimised cutoff does not allow a derivative expansion beyond second order. The fundamentals and applications of the “exact renormalization group” (exact RG), discovered and so christened independently by Wilson and Wegner [1, 2], have been studied intensively since the beginning of the nineties [3–5]. The central reason for this recrudescence is the general acceptance that, far from being merely formal exact realizations of Wilson’s RG ideas, these ideas form the basis for powerful and flexible approximations in non-perturbative quantum field theory. (For reviews, see for example refs. [6].) The two most widely used realizations of such exact RGs (for others see [7, 8]), are Polchinski’s version [9], equivalent to Wilson’s [1] by a change of variables [10, 11], and the version for the Legendre effective action [3–5]. We will be interested in the case where these are applied to O(N) invariant N -component real scalar field theory in D Euclidean dimensions. For such a theory, Polchinski’s version is given by: ∂SΛ ∂Λ = 1 2 δSΛ δΦa · ∂∆UV ∂Λ · δSΛ δΦa − 1 2 tr ∂∆UV ∂Λ · δ SΛ δΦδΦ , (1) where Φa(x) is the N -component scalar field and SΛ[Φ] is the interaction part of the Wilsonian effective action S Λ = 1 2 Φa.∆ −1 UV .Φa + SΛ. (2) Λ is the effective cutoff, ∆UV (q,Λ) = CUV (q,Λ)/q 2 is the ultraviolet regularised propagator, and CUV the ultra-violet cutoff function. On the other hand, the flow equation for the Legendre effective action, also called effective average action, is given by: ∂ ∂Λ ΓΛ[φ] = − 1 2 tr 1 ∆IR ∂∆IR ∂Λ · A (3) where Aab = δab +∆IR · δΓΛ δφaδφb . (4) Here ΓΛ[φ] is the interaction part of the Legendre effective action Γ Λ = 1 2 φa ·∆ IR · φa + ΓΛ[φ], (5) where the propagator has been replaced by an infrared regularised propagator ∆IR(q,Λ) = CIR(q /Λ)/q, CIR being the infrared cutoff function. One of the simplest and most powerful approximations, which is also widely used, is the Local Potential Approximation (LPA) [13]. In this case one simply makes the model approximation that the above actions are of the form of a potential only, and discards all parts of the right hand sides of (1) and (3) that do not fit this approximation. More rigorously, providing the cutoff functions are smooth, the actions have a derivative expansion to all orders [10], and the LPA simply amounts to taking the lowest order in this expansion, setting all higher order terms to zero. We base our notation on refs. [5, 12] for reasons that will become clear.