content_image/413181.jpg content_image/413161.jpg content_image/413160.jpg content_image/413180.jpg content_image/413162.jpg content_image/413179.jpg content_image/413159.jpg content_image/413165.jpg content_image/413158.jpg content_image/413178.jpg
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]: 

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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 

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. 

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 

such that an invariant and quadratic regulator action is the superspace integral of 

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Expressed in component fields, we find 

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In momentum space, i∂$_{µ}$ is replaced by p$_{µ}$ and the oper- ators take the explicit forms 

with 

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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. 

We employ the following ansatz for the supersymmetric effective action for our study of the three-dimensional Wess-Zumino model: 

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 

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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 

## A. Local Potential Approximation 

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