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in great detail and it has been shown that optimized reg- ulator functions minimizing the trucational error can be constructed, see e. g. Refs. [4, 17, 24, 25, 27, 28]. 

Contrary to the model in two dimensions [13], the reg- ulator function (18) regularizes the flow even if we allow for running wave function renormalizations. For the su- perpotential W$_{k}$ we then obtain the simple flow equation 

As we are interested in the bosonic potential V ( φ ) = W ′ $^{2}$( φ ) / 2 we will consider mostly the flow equation for W ′ $_{k}$( φ ) which reads 

Fig. 1 shows the flow of W ′ 2 k ( φ ) for a quadratic super- potential at the cutoff scale, W ′ Λ = λ$_{Λ}$ ( φ 2 − a 2 $_{Λ}$), and with initial conditions λ$_{Λ}$/ Λ = 1, a 2 $_{Λ}$/ Λ 0 . 5 = 0 . 02. With these initial conditions the RG flow starts in the regime with broken Z$_{2}$ symmetry and for k → 0 ends up in the regime with restored Z$_{2}$ symmetry. We observe that the potential V$_{k}$ becomes flat at the origin as k is lowered to the infrared. In addition the function W ′′ k → 0 is regular for all values of the field, in contrast to the situation in two dimensions. 

In order to study the fixed-point structure we introduce dimensionless quantities 

The dimensionless flow equation then reads 

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and its fixed points are characterized by ∂$_{t}$w$_{∗}$ = 0. The flow equations in two and three dimensions have almost identical forms. In three dimensions, however, there ap- pears the additional term ∝ ϕw ′ $_{k}$( ϕ ), since the field φ itself is a dimensionful quantity. 

We observe a further peculiarity of supersymmetric Wess-Zumino models: Only the second derivative of the superpotential enters the fixed-point equation following from ∂$_{t}$w$_{∗}$ = 0, see [13]. It follows that the couplings of the terms φ 0 and φ 1 do not enter the fixed-point equation but evolve independently. As we shall see below, this has some interesting consequences which distinguish the su- persymmetric Wess-Zumino model from purely bosonic theories, for example O ( N ) models in three dimensions, see e. g. [2, 16–18]. 

For our fixed-point analysis, we study the first deriva- tive of Eq. (22), 

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where the prime denotes the derivative with respect to the dimensionless field ϕ . 

First we solve Eq. (23) in the polynomial approxima- tion with a GLYPH<CI> 2 symmetric w ′ Λ at the cutoff scale. The RG flow is such that a GLYPH<CI> 2 symmetric w ′ Λ will remain GLYPH<CI> 2 symmetric during the flow. Thus a polynomial approxi- mation for w ′ Λ is of the form 

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where λ , a 2 and b$_{2}$$_{i}$ denote the scale-dependent couplings. Recall that for an even function w $^{′}$( ϕ ) supersymmetry may be broken. We find the following infinite tower of differential equations: 

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Note that due to supersymmetry the lowest order cou- pling a 2 does not enter the flow equations of the couplings λ, b$_{4}$, b$_{6}$, . . . . 

In our fixed-point analysis we find a Gaußian fixed point with all coupling constants equal to zero and, due to the Z$_{2}$ symmetry, a pair of fixed-points whose cou- plings converge rapidly for larger truncations as shown 

## 1. Polynomial approximation 

