4 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), FIG. 1. RG flow of W ′ 2 k ( φ ) with the initial conditions λ Λ / Λ = 1, a 2 $_{Λ}$/ Λ 0 . 5 = 0 . 02. Note that t is related to k via t = ln k $^{2}$, see also Eq. (10). where the prime denotes the derivative with respect to the dimensionless field ϕ . 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 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 where λ , a 2 and b$_{2}$$_{i}$ denote the scale-dependent couplings. Recall that for an even function w $^{′}$( ϕ ) supersymmetry may be broken. We find the following infinite tower of differential equations: Fig. 1 shows the flow of W ′ 2 k ( φ ) for a quadratic super- potential at the cutoff scale, W ′ Λ = λ$_{Λ}$ ( φ 2 − a 2 $_{Λ}$), and with initial conditions λ$_{Λ}$/ Λ = 1, a 2 $_{Λ}$/ Λ 0 . 5 = 0 . 02. With these initial conditions the RG flow starts in the regime with broken Z$_{2}$ symmetry and for k → 0 ends up in the regime with restored Z$_{2}$ symmetry. We observe that the potential V$_{k}$ becomes flat at the origin as k is lowered to the infrared. In addition the function W ′′ k → 0 is regular for all values of the field, in contrast to the situation in two dimensions. In order to study the fixed-point structure we introduce dimensionless quantities Note that due to supersymmetry the lowest order cou- pling a 2 does not enter the flow equations of the couplings λ, b$_{4}$, b$_{6}$, . . . . In our fixed-point analysis we find a Gaußian fixed point with all coupling constants equal to zero and, due to the Z$_{2}$ symmetry, a pair of fixed-points whose cou- plings converge rapidly for larger truncations as shown The dimensionless flow equation then reads ∂$_{t}$a $^{2}$( t ) = a $^{2}$( t ) ( − 3 λ ( t ) 2 π 2 + 3 b$_{4}$ ( t ) 2 π $^{2}$λ ( t ) − 1 ) + 1 4 π 2 , ∂$_{t}$λ ( t ) = − 3 b$_{4}$ t ) − 6 λ ( t ) 3 + π $^{2}$λ ( t ) 2 π 2 , (25) ∂$_{t}$b$_{4}$ ( t ) = 120 b$_{4}$ ( t ) λ ( t ) 2 + 2 π $^{2}$b$_{4}$ ( t ) − 15 b$_{6}$ ( t ) − 80 λ ( t ) 5 4 π 2 . ϕ = k − 1 2 φ, w$_{k}$ ( ϕ ) = k − $^{2}$W$_{k}$ ( φ ) , w ′ $_{k}$( ϕ ) = k − 3 2 W ′ $_{k}$( φ ) , . . . . (21) ∂$_{t}$w$_{k}$ + 2 w$_{k}$ = − w ′′ k 8 π $^{2}$(1 + w ′′ 2 k ) + ϕw ′ k 2 , (22) w ′ $_{k}$( ϕ ) = λ ( t ) ( ϕ 2 − a $^{2}$( t ) + n ∑ i =2 b$_{2}$$_{i}$ ( t ) ϕ 2 i , (24) ∂$_{k}$W ′ k = − k $^{2}$W (3) k ( φ ) ( k 2 − W ′′ $_{k}$( φ ) 2 ) 8 π $^{2}$( k 2 + W ′′ $_{k}$( φ ) $^{2}$) 2 . (20) ∂$_{k}$W$_{k}$ = − k 2 8 π 2 W ′′ $_{k}$( φ ) k 2 + W ′′ k ( φ ) $_{2}$. (19) ∂$_{t}$w ′ k = ϕw ′′ k − 3 w ′ k 2 + w ′′ 2 k w ′′′ k 4 π $^{2}$(1 + w ′′ 2 k ) 2 − w ′′′ k 8 π $^{2}$(1 + w ′′ 2 k ) , (23)  0  0.0001  0.0002  0.0003  0.0004  0.0005  0.0006  0.0007  0.0008 -0.15 -0.1 -0.05  0  0.05  0.1  0.15 W’ k 2 ( ) t= 0.0 t=-0.1 t=-0.3 t=-0.5 t=- 20 1. Polynomial approximation