3 In momentum space, i∂$_{µ}$ is replaced by p$_{µ}$ and the oper- ators take the explicit forms r$_{1}$ = 0 , r$_{2}$ = ( k | p | − 1 ) θ ( k 2 − p $^{2}$) , (18) ∂$_{k}$W$_{k}$ ( φ ) = 1 2 ∫ d $^{3}$p (2 π ) 3 ∂$_{k}$r$_{1}$ (1 + r$_{2}$ ) − ∂$_{k}$r$_{2}$ ( W ′′ k ( φ ) − r$_{1}$ ) p $^{2}$( r$_{2}$ + 1) 2 + ( W ′′ k ( φ ) + r$_{1}$ ) 2 . (17) Δ S$_{k}$ = 1 2 ∫ ( φ, F ) R B k ( φ F ) + 1 2 ∫ ¯ ψR F $_{k}$ψ. (13) Φ K ( ¯ DD )Φ = Φ ( r$_{1}$ ( − □ ) − r$_{2}$ ( − □ ) ¯ DD 2 ) Φ . (12) ( 1 2 ¯ DD ) 2 n = ( − □ ) n , (11) such that an invariant and quadratic regulator action is the superspace integral of Expressed in component fields, we find Γ$_{k}$ = ∫ d $^{3}$x [ 1 2 ∂$_{µ}$φ∂ $^{µ}$φ − i 2 ¯ ψ / ∂ψ + 1 2 F 2 + F W ′ $_{k}$( φ ) − 1 2 W ′′ $_{k}$( φ ) ¯ ψψ ] . (16) In the following we work in the so-called local potential approximation (LPA) where the expectation values of the fields are taken to be constant over the entire volume. As it has been found in studies of scalar O ( N ) models and low-energy QCD models, the LPA captures already most qualitative and quantitative features associated with crit- ical dynamics at zero and finite temperature provided the anomalous dimensions are small, see e. g. Refs [2, 16– 18, 32, 34]. For the time being we restrict our study to LPA. In Sect. III B we shall then discuss the running of the wave-function renormalization. In order to obtain a flow equation for the superpoten- tial, we project Eq. (10) onto the terms linear in the auxiliary field F and integrate the resulting W ′ k with re- spect to φ . Performing a Wick rotation of the zeroth component of the momentum, i. e. p M 0 → ip E 0 , we find the flow equation In the following we choose the simple regulator functions for which the momentum integration in (17) can be per- formed analytically. In the present work we do not aim at a study the regulator dependence. However, the reg- ulator dependence of functional RG flows, in particular with respect to critical phenomena, has been investigated 2 In the following we neglect an additional term to the flow equa- tion arising from the normalization of the Gaußian measure in the partition function. Including such an additional term yields a field-independent constant to W $^{′}$( φ ). We stress that only non- universal quantities such as the critical temperature for the GLYPH<CI> 2 phase transition are affected by such a constant. However, our analysis of the critical dynamics at the phase boundary at zero and finite temperature is not affected. 3 A Regulator term quadratic in the fluctuation fields ensures that we eventually obtain a non-perturbative RG equation with one- loop structure. We employ the following ansatz for the supersymmetric effective action for our study of the three-dimensional Wess-Zumino model: To construct a supersymmetric flow we use as regulator an invariant D term in superspace. Since such a term should be quadratic in the fields $^{3}$, see App. A for details, it is the D term of a superfield Φ K Φ with K being a function of ¯ DD . Using the anticommutation relation (5), powers of ¯ can always be decomposed into where the indices a, b denote the field components, in- ternal and Lorentz indices, as well as space-time or mo- mentum coordinates, i.e., Ψ T = ( φ, F, ψ, ¯ ψ ) is a vector of fields, not to be confused with a superfield. The cutoff function R$_{k}$ provides an infrared (IR) cut-off for all fields and specifies the Wilsonian momentum-shell integrations such that the flow of Γ$_{k}$ is dominated by modes p 2 ≃ k $^{2}$. For a derivation 2 of the RG flow equation in Minkowski space-time (10) we refer to App. A. where r$_{1}$ = r$_{1}$ ( p $^{2}$/k$^{2}$) and r$_{2}$ = r$_{2}$ ( p $^{2}$/k$^{2}$). Note that the requirement that the RG flow preserves supersym- metry relates the regulators R B k and R F k in the bosonic and fermionic subsectors. where the scale-dependent effective action Γ$_{k}$ interpo- lates between the microscopic (classical) action Γ$_{k}$$_{=Λ}$ = S and the full quantum effective action Γ$_{k}$$_{=0}$ = Γ. The sec- ond functional derivative in Eq. (10) is defined as A. Local Potential Approximation R$_{k}$ = ( R B k 0 0 R F k ) (14) R B k = ( p $^{2}$r$_{2}$ r$_{1}$ r$_{1}$ r$_{2}$ ) , and R F k = − r$_{1}$ − r$_{2}$ / p , (15) ( Γ (2) k ) ab = − → δ δ Ψ$_{a}$ Γ$_{k}$ ← − δ δ Ψ$_{b}$ , ∂$_{t}$ Γ$_{k}$ = i 2 Str [ ( Γ (2) k + R$_{k}$ ) − 1 ∂$_{t}$R$_{k}$ ] , t = ln k 2 , (10) with the one- and two-dimensional Wess-Zumino models [13, 15] to three dimensions. Since there are no Majorana fermions in three-dimensional Euclidean space we begin with a Minkowski spacetime formulation of the Wetterich equation [37, 38]: